978-3-98547-561-2-volume-3

INTERNATIONAL GONGRESS

sciimath> \mathrm{K} K

-stability involves a strengthened notion called reduced uniform K-stability, which matches the reduced coercivity in (1.14) (see

[19, 29, 39]

). Recall that

\tilde{T}

denotes a maximal torus of

Aut (X, L)

, and

{\tilde{N}}_{Q}

is defined similar to (1.3).

Definition 2.3. A polarized manifold

(X, L)

is uniformly K-stable (resp. reduced uniformly K-stable) if there exists

γ > 0

such that any test configuration

(X, L)

satisfies

M^{N A} (X, L) \geq

γ \cdot J^{N A} (X, L) (

resp.

M^{N A} (X, L) \geq γ \cdot inf_{ξ \in {\tilde{N}}_{Q}} J^{N A} (X_{ξ}, L_{ξ}))

Here the twist

(X_{ξ}, L_{ξ})

is introduced by Hisamoto [39]. One way to define it as a test configuration is by resolving the composition of birational morphisms

(X, L) \to (X_{C} =

X \times C, L_{C} = p_{1}^{*} L) \overset{σ_{ξ}}{\to} (X_{C}, L_{C})

where

σ_{ξ}

is the

C^{*}

-action generated by

ξ

. Alternatively, it can be defined in a more general setting of filtrations (see Example 2.8).

2.2. Non-Archimedean pluripotential theory

We discuss how non-Archimedean pluripotential theory as developed by Boucksom-Jonsson can be applied to study K-stability. Corresponding to a regularization result in the complex analytic case, an u.s.c. function

ϕ : X^{N A} \to R \cup {+ \infty}

is called a nonArchimedean psh potential if it is a decreasing limit of a sequence from

H^{N A}

. Denote the space of such functions by

{P S H}^{N A}

. Boucksom-Jonsson introduced the following nonArchimedean version of the finite energy space. First corresponding to (1.10), for any

ϕ \in

{P S H}^{N A}

, define

E^{N A} (ϕ) = inf {E^{N A} (\tilde{ϕ}) : \tilde{ϕ} \geq ϕ, \tilde{ϕ} \in H^{N A}}

Then, corresponding to (1.11), define the space of non-Archimedean finite energy potentials by

{(ε^{1})}^{N A} = {ϕ \in {P S H}^{N A} : E^{N A} (ϕ) > - \infty}

This space is again equipped with a strong topology which makes

E^{N A}

continuous. Boucksom-Jonsson showed in [22] that the non-Archimedean Monge-AmpÃ¨re measure

{M A}^{N A} (ϕ)

is well defined for any

ϕ \in {(E^{1})}^{N A}

such that if

{ϕ_{k}}_{k \in N} \subset H^{N A}

converges to

ϕ

strongly, then

{M A}^{N A} (ϕ_{k})

converges to

{M A}^{N A} (ϕ)

weakly.

A large class of potentials come from filtrations (see [19]). Set

R_{m} = H^{0} (X, m L)

Definition 2.4. A filtration is the data

F = {F^{λ} R_{m} \subseteq R_{m}; λ \in R, m \in N}

that satisfies the following four conditions:

(i)

F^{λ} R_{m} \subseteq F^{λ^{'}} R_{m}

, if

λ \geq λ^{'}

; (ii)

F^{λ} R_{m} = ⋂_{λ^{'} < λ} F^{λ^{'}} R_{m}

;

(iii)

F^{λ} R_{m} \cdot F^{λ^{'}} R_{m^{'}} \subseteq F^{λ + λ^{'}} R_{m + m^{'}}

, for

λ, λ^{'} \in R

and

m, m^{'} \in N

;

(iv) There exist

e_{-}, e_{+} \in Z

such that

F^{m e}_{-} R_{m} = R_{m}

and

F^{m e_{+}} R_{m} = 0

for

m \in

Z_{\geq 0}

Filtration

F

is finitely generated if its extended Rees algebra

R (F)

is finitely generated where

R (F) = ⨁_{λ \in R} ⨁_{m \in N} t^{- λ} F^{λ} R_{m}

In this case

F

induces a degeneration of

X

into

X_{0} = Proj (⨁_{m, λ} F^{λ} R_{m} / F > λ R_{m})

For a general

F, {F λ R_{ℓ}; λ \in R}

generates a filtration

F (ℓ)

R^{(ℓ)} := ⨁_{m \in N} R_{m ℓ}

, which induces a non-Archimedean psh potential

{\overset{ˇ}{ϕ}}^{(ℓ)} \in H^{N A}

. Define

ϕ_{F} = {(\underset{ℓ \to + \infty}{lim sup} {\overset{ˇ}{ϕ}}^{(ℓ)})}^{*}

where

(\cdot)^{*}

denotes the upper-semicontinuous regularization.

Example 2.5. Filtration

F

is a

Z

-filtration if

F^{λ} R_{m} = F ⌈ λ ⌉ R_{m}

. By

[19, 63, 69]

, there is a one-to-one correspondence between test configurations equipped with relatively ample

Q

polarizations and finitely generated

Z

-filtrations. Any test configuration

(X, L)

defines such a filtration by

Conversely, if

F

is a finitely generated

Z

-filtration, then

(X := {Proj}_{C [t]} (R (\overset{ˇ}{F} (ℓ))), \frac{1}{ℓ} O_{X} (1))

is a test configuration for

ℓ

sufficiently divisible.

Example 2.6. In Definition 2.1 of test configurations, if we do not require

L

to be

π

semiample, then we call

(X, L)

a model (of

(X \times C, p_{1}^{*} L)

). The same definition in (2.7) defines a filtration also denoted by

F_{(x, L)}

. However, in general the filtration is not finitely generated anymore. Fix any model

(X, L)

such that

\bar{L}

is big over

\bar{X}

(we call such

(X, L)

a big model for

(X, L)

). In [46] we obtained the following formula for the non-Archimedean Monge-AmpÃ¨re measure of

ϕ = ϕ_{(X, L)} := ϕ_{F_{(X, L)}}

which generalizes (2.5):

\begin{matrix} (2.8) & {MA}^{N A} (ϕ) = \sum_{i} b_{i} (⟨ L^{\cdot n} ⟩ \cdot F_{i}) δ_{v_{i}} \end{matrix}

Here for any divisor

D

, we use the notion of a positive intersection product introduced in [17]:

⟨ {\bar{L}}^{n + 1} ⟩ := vol (\bar{L}) = lim_{m \to + \infty} \frac{h^{0} (\bar{X}, m \bar{L})}{\frac{m^{n + 1}}{(n + 1)!}}, ⟨ {\bar{L}}^{n} ⟩ \cdot D := {\frac{1}{n + 1} \frac{d}{d t} |}_{t = 0} vol (\bar{L} + t D)

Example 2.7. Any

v \in X_{Q}^{div}

defines a filtration: for any

λ \in R

and

m \in Z_{\geq 0}

, define

\begin{matrix} (2.9) & F_{v}^{λ} R_{m} = {s \in R_{m} : v (s) \geq λ} \end{matrix}

Boucksom-Jonsson proved in [21] that

{M A}^{N A} (ϕ_{F_{v}}) = V \cdot δ_{v}

Example 2.8. Assume a torus

\tilde{T} ≅ {(C^{*})}^{r}

-acts on

(X, L)

. Then we have a weight decomposition

R_{m} = ⨁_{α \in Z^{r}} R_{m, α}

. For any

ξ \in {\tilde{N}}_{R}

, we can define the

ξ

-twist of a given filtration:

F_{ξ}^{λ} R_{m} = F^{λ - ⟨ α, ξ ⟩} R_{m, α}

. On the other hand, there is an induced

{\tilde{N}}_{R}

-action on

{(X^{N A})}^{\tilde{T}}

which sends

(ξ, v)

v_{ξ} \in {(X^{N A})}^{\tilde{T}}

determined by the following condition: if

f \in C (X)_{α}

which means

f \circ t^{- 1} = t^{α} \cdot f

for any

t \in \tilde{T}

, then

v_{ξ} (f) = ⟨ α, ξ ⟩ + v (f)

. We then have the following formula:

{M A}^{N A} (ϕ_{F_{ξ}}) = (- ξ)_{*} {M A}^{N A} (ϕ_{F})

(see

[44, 45]

Generalizing the case of test configurations, Boucksom-Jonsson showed that the non-Archimedean functionals from (2.2)-(2.4) are well defined for all

ϕ \in {(E^{1})}^{N A}

by using integrals over

X^{N A}

mentioned before (for example, for

H^{N A}

use (2.6)).

Example 2.9. For any filtration

F

, it is known that

ϕ_{F} \in {(E^{1})}^{N A}

. Following [19], define

vol (F^{(t)}) = lim_{m \to + \infty} \frac{\dim_{C} F^{m t} R_{m}}{m^{n} / n!}

Then

E^{N A}

is the following "expected vanishing order" with respect to

F

(see [21]).

\begin{matrix} (2.10) & E^{N A} (ϕ_{F}) = \frac{1}{V} \int_{R} t (- d vol (F^{(t)})) \end{matrix}

Similar to Theorem 1.2, we also have important regularization properties:

Theorem 2.10 ([22]). For any

ϕ \in {(E^{1})}^{N A}

, there exists

{ϕ_{k}}_{k \in N} \subset H^{N A}

(i.e.,

ϕ_{k} = ϕ_{(X_{k}, L_{k})}

for a test configuration

(X_{k}, L_{k}))

such that

ϕ_{k} \to ϕ

in the strong topology and

F^{N A} (ϕ_{k}) \to

F^{N A} (ϕ)

for

F \in {E, Λ, E^{K_{X}}}

Boucksom-Jonsson conjectured that the same conclusion should also hold for

H^{N A}

. This conjecture is still open in general and it is important in the non-Archimedean approach to the YTD conjecture. We have made progress in this direction.

Theorem 2.11 ([45,46]). (1) For any

ϕ \in {(E^{1})}^{N A}

, there exist models

{(X_{k}, L_{k})}_{k \in N}

such that

ϕ_{k} = ϕ_{(x_{k}, L_{k})} \to ϕ

in the strong topology and

H^{N A} (ϕ_{k}) \to H^{N A} (ϕ)

(2) For any big model

(X, L)

, we have the following formula that generalizes (2.4):

M^{N A} (X, L) = \frac{1}{V} ⟨ L^{\cdot n} ⟩ \cdot K_{\bar{x} / P^{1}} + \frac{\underline{S}}{(n + 1) V} ⟨ L^{\cdot n + 1} ⟩

The idea for proving the first statement is similar to the Archimedean setting in [8]. First we regularize the measure

{M A}^{N A} (ϕ)

with converging entropy. In fact, we find a way to regularize it by using measures supported at finitely many points in

X_{Q}^{div}

. Then we use the solution of non-Archimedean Monge-AmpÃ¨re equations obtained in [18] to get the wanted potentials which are known to be associated to models. However, in the non-Archimedean case, there is not yet a characterization of measures associated to test configurations which prevents us from regularizing via test configurations. The second statement in Theorem 2.11 follows from the formula (2.8), and it prompts us to propose the following algebro-geometric conjecture which would strengthen the classical Fujita approximation theorem.

Conjecture 2.12. Let

\bar{X}

be a smooth

(n + 1)

-dimensional projective variety. Let

\bar{L}

be a big line bundle over

\bar{X}

. Then there exist birational morphisms

μ_{k} : {\bar{X}}_{k} \to \bar{X}

and decompositions

μ_{k}^{*} \bar{L} = {\bar{L}}_{k} + E_{k}

N^{1} (\bar{X})_{Q}

with

{\bar{L}}_{k}

semiample and

E_{k}

effective such that

lim_{k \to + \infty} {\bar{L}}_{k}^{n + 1} = vol (\bar{L}), lim_{k \to + \infty} {\bar{L}}_{k}^{n} \cdot K {\bar{x}}_{k} = {\frac{1}{n + 1} \frac{d}{d t} vol (\bar{L} + t K \bar{x}) |}_{t = 0} = ⟨ {\bar{L}}^{\cdot n} ⟩ \cdot K \bar{x}

It is easy to show that this conjecture is true if

\bar{L}

admits a birational Zariski decomposition. The author verified this conjecture for certain examples of big line bundles due to Nakamaya which do not admit such decompositions (see [46]). Y. Odaka observed that when

(X, L)

is a big model for a polarized spherical manifold (for example, a polarized toric manifold),

\bar{X}

is a Mori dream space which implies that

\bar{L}

admits a Zariski decomposition and hence the above conjecture holds true.

2.3. Stability of Fano varieties

In this section, we assume that

X

is a

Q

-Fano variety (i.e.,

- K_{X}

is an ample

Q

line bundle and

X

has at worst klt singularities). Corresponding to (1.9), we have a nonArchimedean D functional. For general test configurations, it first appeared in Berman's work [4] and was reformulated in [19] using non-Archimedean potentials:

L^{N A} (X, L) = inf_{v \in X_{Q}^{d i v}} (A_{X} (v) + ϕ_{(X, L)} (v)), D^{N A} (X, L) = - E^{N A} (X, L) + L^{N A} (X, L)

The notions of Ding-stability and uniform Ding-stability are defined if

M^{N A}

is replaced by

D^{N A}

in Definitions 2.2 and 2.3. In general, we have the inequality

M^{N A} (X, L) \geq D^{N A} (X, L)

. For Fano varieties, special test configurations play important roles. A test configuration

(X, L)

is called special if the central fiber

X_{0}

is a

Q

-Fano variety and

L = - K_{X / P^{1}}

. For special test configurations, we have

D^{N A} = M^{N A} = - E^{N A} =

: Fut

x_{0} (ξ)

, the last quantity being the Futaki invariant on

X_{0}

for the holomorphic vector field

ξ

that generates the

C^{*}

-action. The importance of special test configurations was first pointed out in Tian's work [64] motivated by compactness results from metric geometry. The following results show their importance from the point of view of algebraic geometry:

Theorem 2.13 ([35,44,52], see also

[7, 21])

. For any

Q

-Fano variety,

K

-stability is equivalent to Ding-stability, and they are equivalent to

K

-stability or Ding-stability over special test configurations. Moreover, the same conclusion holds true if stability is replaced by semistability, polystability, or reduced uniform stability.

The proofs of these results depend on a careful process of Minimal Model Program first used in [52] to transform any given test configuration into a special one. Moreover, crucial calculations show that the relevant invariants such as

M^{N A}

D^{N A}

decrease along the MMP process. Theorem 2.13 leads directly to a valuative criterion for K-stability. To state it, first define for any

v \in X_{Q}^{div}

an invariant (see Example 2.9):

\begin{matrix} (2.11) & S_{L} (v) := \frac{1}{V} \int_{0}^{+ \infty} vol (F_{v}^{(t)}) d t = \frac{1}{V} \int_{R} t (- d vol (F_{v}^{(t)})) = E^{N A} (ϕ_{F_{v}}) \end{matrix}

Let

\tilde{T}

be a maximal torus of

Aut (X)

and

{(X_{Q}^{div})}^{\tilde{T}}

be the set of

\tilde{T}

-invariant divisorial valuations. Define the following invariant

((ξ, v) \mapsto v_{ξ}

is the action appeared in Example 2.8):

δ (X) = inf_{v \in X_{Q}^{d i v}} \frac{A_{X} (v)}{S_{X} (v)}, δ_{\tilde{T}} (X) = inf_{v \in (X_{Q}^{d i v})} sup_{ξ \in {\tilde{N}}_{R}} \frac{A_{X} (v_{ξ})}{S_{X} (v_{ξ})}

Here we use the convention that

A_{X} (v_{triv}) / S_{X} (v_{triv}) = + \infty

for the trivial valuation

v_{triv}

Theorem 2.14. The following statements are true.

(1)

([35, 42]) X

K

-semistable if

δ (X) \geq 1

(2)

([34, 35]) X

is uniformly

K

-stable if and only if

δ (X) > 1

(3) (

[15, 35, 42]) X

K

-stable if and only if

A_{X} (v) > S (v)

for any nontrivial

v \in X_{Q}^{div}

(4) ([44])

X

is reduced uniformly

K

-stable if and only if

δ_{\tilde{T}} (X) > 1

To get these, we first use the fact as pointed out in [19] that for a special test configuration

(X, L)

, the valuation ord

X_{0}

of the function field

C (X \times C)

restricts to become a divisorial valuation

v \in X_{Q}^{div}

. A crucial observation is then made in [42]:

F_{(x, L)}^{λ} R_{m} =

F_{v}^{λ + m A_{X} (v)} R_{m} (

see (2.9)). This implies

vol (F_{(X, L)}^{(t)}) = vol (F_{v}^{(t + A_{X} (v))})

, which, together with (2.10), leads to

^{2}

\begin{matrix} (2.12) & M^{N A} (X, L) = D^{N A} (X, L) = A_{X} (v) - E^{N A} (F_{v}) = A_{X} (v) - S (v) \end{matrix}

This, together with Theorem 2.13, already gives the sufficient condition for the K-(semi)stability. The criterion for uniform K-stability follows from a similar argument and K. Fujita's inequality,

\frac{1}{n} S (v) \leq J^{N A} (F_{v}) \leq n S (v)

[34]. For reduced uniform stability, another identity

A_{X} (v_{ξ}) - S (v_{ξ}) = A_{X} (v) - S (v) +

Fut

_{X} (ξ)

proved in [44] is needed.

As we will see in Section 3.2, a main reason for introducing the (reduced) uniform K-stability is that it is much easier to use in making connection with the (reduced) coercivity in the complex analytic setting. On the other hand, we now have the following fundamental result:

Theorem 2.15 ([56]). Let

X

be a

Q

-Fano variety. Then

X

K

-stable if and only if

X

is uniformly

K

-stable. More generally,

X

is reduced uniformly stable if and only if

X

K

polystable. Moreover, these statements hold true for any log Fano pair.

This is achieved by several works. First, according to a work of Blum-Liu-Xu [13], divisorial valuations on

X

associated to special test configurations are log canonical places of complements. By deep boundedness of Birkar and Haccon-McKernan-Xu, it was also shown that there exists a quasimonomial valuation (i.e., a monomial valuation on a smooth birational model) that achieves the infimum defining

δ (X)

(and a similar result holds more generally for

δ_{\tilde{T}} (X))

. Then the main problem becomes proving a finite generation property for the minimizing valuation, which is achieved by using deep techniques from birational

2 The original argument in [42] also explicitly relates the filtration

F_{(} X, L)

to a filtration of the section ring of

X_{0}

induced by the

C^{*}

-action.
algebraic geometry in [56]. In fact, in the past several years, the algebraic study of K-stability for Fano varieties has flourished, and there are many important results which answer fundamental questions in this subject. We highlight two such achievements:

(1) Algebraic construction of projective moduli space of K-polystable Fano varieties. This is achieved in a collection of works, settling different issues in the construction including boundedness, separatedness, properness, and projectivity. Moreover, concrete examples of compact moduli spaces have been identified. We refer to

[56, 70]

for extensive discussions on related topics.

(2) Fujita-Odaka [36] introduced quantizations of the

δ (X)

invariant: for each

m \in N

δ_{m} (X) = inf {lct (X, D) : D is of m -basis type}

where

D

is of

m

-basis type if

D = \frac{1}{m N_{m}} \sum_{i = 1}^{N_{m}} {s_{i} = 0}

where

{s_{i}}

is a basis of

H^{0} (X, m L)

. Blum-Jonsson [12] proved

lim_{m \to + \infty} δ_{m} (X) = δ (X)

. This provides a practical tool to verify uniform stability of Fano varieties. Ahmadinezhad-Zhuang [1] further introduced new techniques for estimating the

δ_{m}

and

δ

invariant which lead to many new examples of K-stable Fano varieties. All of these culminate in the recent determination of deformation types of smooth Fano threefolds that contain K-polystable ones (see [3]).

In another direction, Han-Li [37] establishes a valuative criterion for

g

-weighted stability, corresponding to the study of

g

-solitons. A key idea in such an extension is using a fibration technique for a polynomial weights (as motivated by the theory of equivariant de Rham cohomology) and then using the Stone-Weierstrass approximation to deal with the general

g

. Moreover, there is a notion of stability for Fano cones introduced earlier by Collins-SzÃ©kelyhidi associated to Ricci-flat KÃ¤hler cone metrics. It is shown recently that this stability of Fano cones is, in fact, equivalent to a particular

g

-weighted stability of log Fano quotients (see

[2, 47]

The techniques developed in the study of (weighted) K-stability of Fano varieties have also been applied to treat an optimal degeneration problem that is motivated by the Hamilton-Tian conjecture in differential geometry (see [74] for background of this conjecture). This is formulated as a minimization problem for valuations in [38] which defines the following functional (cf. (2.12) and (2.11)), for any valuation

v \in X^{N A}

\tilde{β} (v) = A_{X} (v) + \log (\frac{1}{V} \int_{0}^{+ \infty} e^{- t} (- d vol (F_{v}^{(t)})))

Very roughly speaking, the

\tilde{β}

functional is an antiderivative of a certain weighted Futaki invariant. This functional is a variant of invariants that appeared in previous works of TianZhang-Zhang-Zhu, Dervan-SzÃ©kelyhidi, and Hisamoto (see [74] for more details). The results from

[14, 37, 51]

together prove the following algebraic version of Hamilton-Tian conjecture:

Theorem 2.16. For any

Q

-Fano variety, there exists a unique quasimonomial valuation

v_{*}

that minimizes

\tilde{β}

, whose associated filtration

F_{v_{*}}

is finitely generated and induces a degener-
ation of

X

to a

Q

-Fano variety

X_{0}

together with a vector field

V_{ξ}

. Moreover,

X_{0}

degenerates uniquely to an

e^{- ⟨ \cdot, ξ ⟩}

-weighted polystable

Q

-Fano variety (cf. Example 1.1).

Combined with previous works, the uniqueness part in particular confirms a conjecture of Chen-Sun-Wang about the algebraic uniqueness of limits under normalized KÃ¤hlerRicci flows on Fano manifolds (see [62]).

2.4. Normalized volume and local stability theory of klt singularities

A similar minimization problem for valuations was actually studied even earlier in the local setting, which motivates the formulation and the proof of Theorem 2.16. Let

(X, x)

be a klt singularity. Denote by

{V a l}_{X, x}

the space of real valuations that have center

x

. The following normalized volume functional was introduced in [43]: for any

v \in {Val}_{X, x}

\begin{matrix} (2.13) & \hat{vol} (v) := {\begin{cases} A_{X} (v)^{n} \cdot vol (v), & if A_{X} (v) < + \infty \\ + \infty, & otherwise \end{cases} \end{matrix}

Here

A_{X} (v)

is again the

\log

discrepancy functional and

vol (v)

is defined as

vol (v) = lim_{p \to + \infty} \frac{\dim_{C} (O_{X, x} / a_{p} (v))}{p^{n} / n!} where a_{p} (v) = {f \in O_{X, x}; v (f) \geq p}

The expression in (2.13) is inspired by the work of Martelli-Sparks-Yau [57] on a volume minimization property of Reeb vector fields associated to Ricci-flat KÃ¤hler cone metrics. In [43] we started to consider the minimization of

\hat{vol over} {Val}_{X, x}

and define the invariant

\hat{vol} (X, x) = inf_{v \in {V a l}_{X, x}} \hat{vol} (v)

. We proved that the invariant

\hat{vol} (X, x)

is strictly positive and further conjectured the existence, uniqueness of minimizing valuations which should have finite generated associated graded rings. For a concrete example, it was shown by the author and Y. Liu that for an isolated quotient singularity

X = C^{n} / Γ, \hat{vol} (C^{n} / Γ, 0) = \frac{n^{n}}{| Γ |}

and the exceptional divisor of the standard blowup obtains the infimum.

This minimization problem was proposed by the author to attack a conjecture of Donaldson-Sun, which states that the metric tangent cone at any point on a GromovHausdorff limit of KÃ¤hler-Einstein manifolds depends only on the algebraic structure (see [62]). This conjecture has been confirmed in a series of following-up papers [51, 53, 54]. Algebraically, we have the following results regarding this minimization problem.

Theorem 2.17. (1) There exists a valuation that achieves the infimum in defining

\hat{vol} (X, x)

. Moreover, this minimizing valuation is quasimonomial and unique up to rescaling.

(2) A divisorial valuation

v_{*}

is the minimizer if and only if it is the exceptional divisor of a plt blowup and also the associated log Fano pair is

K

-semistable.

The first statement is a combination of works by Harold Blum, Chenyang Xu, and Ziquan Zhuang [11,71,72] partly based on calculations from [42,43]. The second statement was proved in

L i - X u

[54] (see also [11]) by extending the global argument from [52] to the local case, and it shows a close relationship between the local and global theory. In fact, it is in proving the affine cone case of this statement when valuative criterion for

K

-(semi)stability
was first discovered in [42]. A similar statement is true for more general quasimonomial minimizing valuations [53]. However, the finite generation conjecture from [43] is still open in general, and seems to require deeper boundedness property of Fano varieties.

We also like to mention that Yuchen Liu obtained a surprising local-to-global comparison inequality by generalizing an estimate of Kento Fujita:

Theorem 2.18 ([55]). For any closed point

x

on a

K

-semistable

Q

-Fano variety

X

, we have

\begin{matrix} (2.14) & {(- K_{X})}^{\cdot n} \leq \frac{(n + 1)^{n}}{n^{n}} \hat{vol} (X, x) \end{matrix}

For example, if

x \in X

is a regular point, (2.14) recovers Fujita's beautiful inequality, namely

{(- K_{X})}^{\cdot n} \leq (n + 1)^{n}

for any K-semistable

X

[33]. Inequality (2.14) has applications in controlling singularities on the varieties that correspond to boundary points of moduli spaces. In order for this to be effective, good estimates of

\hat{vol} (X, x)

for klt singularities need to be developed. In particular, it is still interesting to understand better the

\hat{vol invariants and}

associated minimizers for 3-dimensional klt singularities. For more discussion on related topics, we refer to the survey [48].

3. ARCHIMEDEAN (COMPLEX ANALYTIC) THEORY VS. NON-ARCHIMEDEAN THEORY

3.1. Correspondence between Archimedean and non-Archimedean objects

In this section, we discuss results showing a general philosophy that non-Archimedean objects encode the information of corresponding Archimedean objects "at infinity."

Let

(X, L)

be a test configuration and

\tilde{h}

be a smooth psh metric on

L

. Via the isomorphism

(X, L) \times_{C} C^{*} ≅ X \times C^{*}

, we get a path

\tilde{Φ} = {\tilde{φ} (s)}_{s \in R}

of smooth

ω_{0}

-psh potentials where

s = - \log | t |^{2}

. With these notation, we have the following slope formula:

Theorem 3.1 ([20,59,64,67]). The slope at infinity of a functional

F \in {E, Λ, I, J, M}

is given by the corresponding non-Archimedean functional

F^{' \infty} (\tilde{Φ}) := lim_{s \to + \infty} \frac{F (\tilde{φ} (s))}{s} = F^{N A} (X, L) = F^{N A} (ϕ_{(x, L)})

There is a more canonical analytic object associated to a test configuration. Recall from section 1.5 that by a geodesic ray

Φ = {φ (s)}_{s \in [0, + \infty)}

E^{1}

we mean that

{Φ |}_{[s_{1}, s_{2}]}

is a geodesic connecting

φ (s_{1}), φ (s_{2})

for any

s_{1}, s_{2} \in [0, \infty)

(see (1.12)).

Theorem 3.2 ([60]). Given any test configuration

(X, L)

for

(X, L)

, there exists a geodesic ray

Φ_{(X, L)}

emanating from any given smooth potential

φ_{0}

On the other hand, recall from section 2.1 that there is a non-Archimedean potential associated to

(X, L)

(see (2.1)). Berman-Boucksom-Jonsson proved that there is a direct relation between geodesic rays and non-Archimedean potentials. First they showed that any geodesic ray

Φ

defines a non-Archimedean potential (cf. (2.1))

Φ_{N A} (v) := - G (v) (Φ), for any v \in X_{Q}^{d i v}

where

G (v) (Φ)

is the generic Lelong number of

Φ

considered as a singular quasi-psh potential on a birational model where the center of the valuation

G (v)

is a prime divisor.

Theorem 3.3 ([7]). The following statements are true:

(1) The map

Φ \mapsto Φ_{N A}

has the image contained in

{(E^{1})}^{N A}

. Conversely, for any

ϕ \in

{(ε^{1})}^{N A}

, there exists a geodesic ray denoted by

γ (ϕ)

that satisfies

γ (ϕ)_{N A} = ϕ

(2) For any geodesic ray

Φ, \hat{Φ} = γ (Φ_{N A})

satisfies

{\hat{Φ}}_{N A} = Φ_{N A} \in {(E^{1})}^{N A}

and

\hat{Φ} \geq Φ

(3) For

Φ = γ (ϕ)

with

ϕ \in {(ε^{1})}^{N A}, E^{' \infty} (Φ) = E^{N A} (ϕ)

, and there exists a sequence of test configurations

(X_{m}, L_{m})

such that

Φ

is the decreasing limit of

Φ_{(X_{m}, L_{m})}

(see Theorem 3.2).

Berman-Boucksom-Jonsson proved

Φ_{N A} \in {P S H}^{N A}

by blowing up multiplier ideal sheaves

{L (m Φ)}_{m \in N}

and using their global generation properties to construct test configurations

{(X_{m}, L_{m})}

such that

ϕ_{(X_{m}, L_{m})}

decreases to

Φ_{N A}

. Because of the second statement, any geodesic ray

γ (ϕ)

with

ϕ \in {(E^{1})}^{N A}

is called maximal in [7]. By the last statement, maximal geodesic rays can be approximated by (geodesic rays associated to) test configurations. Moreover, when

ϕ = ϕ_{(x, L)} \in H^{N A}, γ (ϕ)

coincides with the geodesic ray from Theorem 3.2. Further useful properties of maximal geodesic rays are known (cf. Theorem 3.1):

Theorem 3.4 ([45]). Let

Φ

be a maximal geodesic ray.

(1) We have the identity

{(E^{- Ric (ω_{0})})}^{' \infty} (Φ) = {(E^{K})}^{N A} (Φ_{N A})

(2)

H^{' \infty} (Φ) \geq H^{N A} (Φ_{N A})

. Moreover, if

Φ = Φ_{(x, L)}

is associated to a test configuration, then

H^{' \infty} (Φ) = H^{N A} (Φ_{N A})

It is natural to conjecture that

H^{' \infty} (Φ) = H^{N A} (Φ_{N A})

always holds for any maximal geodesic ray

Φ

. This is implied by the algebraic Conjecture 2.12, according to [45, 46].

As pointed out in [7], by a construction of Darvas, there are abundant nonmaximal geodesic rays. In fact, analogous local examples have been used by the author to disprove a conjecture of Demailly on Monge-AmpÃ¨re mass of psh singularities. It is thus a surprising fact that maximal geodesic rays are the only ones of interest in the

c s c K

problem.

Theorem 3.5 ([45]). If a geodesic ray

Φ

satisfies

M^{' \infty} (Φ) < + \infty

, then

Φ

is maximal.

Note that

M^{' \infty} (Φ) = lim_{s \to + \infty} \frac{M (φ (s))}{s}

exists by Theorem 1.3. This result resolves a difficulty raised in Boucksom's ICM talk [16], and implies that destabilizing geodesic rays can always be approximated by test configurations, thus giving a very strong evidence for the validity of Yau-Tian-Donaldson Conjecture 3.6. The proof of Theorem 3.5 starts with an equisingularity property

\int_{X \times {| t | < 1}} e^{- α (\hat{Φ} - Φ)} < + \infty

for any

α > 0

, and then uses Jensen's inequality, together with a comparison principle, for the

E

functional to get a contradiction with the finite slope assumption if

\hat{Φ} = γ (Φ_{N A}) \neq Φ

3.2. Yau-Tian-Donaldson conjecture for general polarized manifolds

The Yau-Tian-Donaldson (YTD) conjecture says that the existence of canonical KÃ¤hler metrics on projective manifolds should be equivalent to a certain K-stability condition. For a general polarization, it is believed that one needs to use a strengthened definition of K-stability such as Definition 2.3. In particular, we have the following version.

Conjecture 3.6 (uniform YTD conjecture). A polarized manifold

(X, L)

admits a cscK metric if and only if

(X, L)

is reduced uniformly

K

-stable.

The implication from existence to stability is known, and follows from Theorems 1.4 and 3.1. The other direction is still open in general. However, based on the results discussed thus far, we can explain the proof of a weak version.

Theorem 3.7 ([45]). If

(X, L)

is uniformly stable over models (i.e., there exists

γ > 0

such that

M^{N A} (X, L) \geq γ \cdot J^{N A} (X, L)

for any model

(X, L))

, then it admits a cscK metric.

Summary of proof. Step 1. By Theorem 1.4, we need to show that

M

is coercive. Assume that the coercivity fails. Then there exists a geodesic ray

Φ = {φ (s)}_{s \in [0, \infty)}

satisfying

M^{' \infty} (Φ) \leq 0, J^{' \infty} (Φ) = 1, sup (φ (s)) = 0

Such a destabilizing geodesic ray

Φ

was constructed in

[7, 27]

from a destabilizing sequence. In this construction, both the convexity of

M

from Theorem 1.3 and a compactness result for potentials with uniform entropy bounds from [6] play crucial roles.

Step 2. By Theorem 3.5,

Φ

is maximal. Set

ϕ = Φ_{N A}

. By using Theorem 3.3(3) and Theorem 3.4(1), we derive the identities

E^{' \infty} (Φ) = E^{N A} (ϕ), {(E^{- Ric (ω_{0})})}^{' \infty} (Φ) = {(E^{K_{X}})}^{N A} (ϕ), J^{' \infty} (Φ) = J^{N A} (ϕ)

Moreover, by Theorem 3.4(2),

H^{' \infty} (Φ) \geq H^{N A} (ϕ)

so that

M^{' \infty} (Φ) \geq M^{N A} (ϕ)

Step 3. By Theorem 2.11, there exist models

(X_{m}, L_{m})

such that

ϕ_{m} = ϕ_{(X_{m}, L_{m})}

converges to

ϕ

in the strong topology and

lim_{m \to + \infty} M^{N A} (ϕ_{m}) = M^{N A} (ϕ), lim_{m \to + \infty} J^{N A} (ϕ_{m}) = J^{N A} (ϕ)

Step 4. We can complete the proof by getting a contradiction to the stability assumption:

0 \geq M^{' \infty} (Φ) \geq M^{N A} (ϕ) = lim_{m \to + \infty} M^{N A} (ϕ_{m}) \geq_{stability} lim_{m \to + \infty} J^{N A} (ϕ_{m}) = J^{N A} (ϕ) = 1

There is a version of Theorem 3.7 in [45] when

Aut (X, L)

is continuous. Moreover, it is shown in [46] that Conjecture 2.12 implies Conjecture 3.6. As mentioned earlier, if (

X, L

) is any polarized spherical manifold, Conjecture 2.12 is true and hence in this case the YTD Conjecture 3.6 is proved. Based on this fact, Delcroix [28] obtained further refined existence results in this case.

We should mention that Sean Paul (see [58]) has works that give a beautiful interpretation of the coercivity of

M

-functional using a new notion of stability for pairs. However, it is not clear how K-stability discussed here can directly imply his stability notion.

3.3. YTD conjecture for Fano varieties

3.3.1. Non-Archimedean approach

Our proof of Theorem 3.7 is in fact modeled on a non-Archimedean approach to the uniform YTD conjecture proposed by Berman-Boucksom-Jonsson in [7]. They carried it out sucessfully for smooth Fano manifolds with discrete automorphism groups. The main advantage in the Fano case is that

D^{N A}

satisfies a regularization property and can be used in place of

M^{N A}

to complete the argument. Recently their work has been extended to the most general setting of

\log

Fano pairs.

Theorem 3.8 ([44, 49,50]). A log Fano pair

(X, D)

admits a KÃ¤hler-Einstein metric if and only if it is reduced uniformly stable for all special test configurations.

Note that this combined with Theorem 2.15 also proves the K-polystable version of the YTD conjecture. Theorem 3.8 can be used to get examples of KÃ¤hler-Einstein metrics on Fano varieties with large symmetry groups (see, for example, [40]). The proof of Theorem 3.8 is much more technical than [7] because we need to overcome the difficulties caused by singularities. The first key idea is to use an approximation approach initiated in [49]. Consider the log resolution

μ : X^{'} \to X

as in Section 1.3 and reorganize (1.4) as

- K_{X^{'}} - D_{ε} = \frac{1}{1 + ε} (μ^{*} (- K_{X} - D) + ε H) =: L_{ε}

where

H = μ^{*} (- K_{X} - D) - \sum_{k} θ_{k} E_{k}

is ample by choosing appropriate

{θ_{k}}

and

D_{ε} =

\sum_{k} (- a_{k} + \frac{ε}{1 + ε}) θ_{k} E_{k}

with

0 \leq ε ≪ 1

. In [49] we considered the simple case when

a_{k} \in

(- 1, 0]

for all

k

. In this case for

0 < ε ≪ 1, (X^{'}, D_{ε})

is a smooth log Fano pair. A crucial calculation using the valuative criterion from Theorem 2.14 shows that (semi)stability of

(X, D)

implies the uniform stability of

(X^{'}, D_{ε})

for

ε > 0

. Moreover, we can prove a version of YTD conjecture for

(X^{'}, D_{ε})

and deduce that it admits a KÃ¤hler-Einstein metric. Next we take a limit as

ε \to 0

to get a KÃ¤hler-Einstein metric on

(X, D)

itself. The proof of this convergence depends on technical uniform pluripotential and geometric estimates.

In [50], we dealt with the general case when

D_{ε}

is not necessarily effective. A key difficulty for the argument in [7] to work on singular varieties is that it is not clear how to use multiplier ideal sheaves to approximate a destabilizing geodesic ray

Φ

when

X

is singular. To circumvent this difficulty, we first need to perturb

Φ

to become a singular quasipsh potential

Φ_{ε}

(X^{'} \times C, p_{1}^{' *} L_{ε})

. Since

X^{'}

is smooth, we know how to approximate

Φ_{ε}

by test configurations for

(X^{'}, L_{ε})

thanks to [7]. However, due to the ineffectiveness of

D_{ε}

, the remaining arguments depend more heavily on non-Archimedean analysis and some key observation on convergence of slopes. In [45] we further derived the valuative criterion for reduced uniform stability and understood how the torus action induces an action on the space of non-Archimedean potentials in order to incorporate group actions in the argument. Note that the non-Archimedean approach a priori does not prove the statement in Theorem 3.8 involving special test configurations. Fortunately, Theorem 2.13 fills this gap.

By using the fibration and approximation techniques mentioned earlier, Theorem 3.8 has been extended to the case of

g

-soliton on log Fano pairs in [38]. As explained in [2, 47], this can be used prove the YTD conjecture for Ricci-flat KÃ¤hler cone metrics thanks to its
equivalence to particular

g

-solitons (see Section 1.4). This generalizes the result of CollinsSzÃ©kelyhidi on YTD conjecture for Fano cones with isolated singularities [25].

3.3.2. Other approaches

For completeness, we briefly mention other approaches to the YTD conjecture on Fano manifolds. The classical way to solve the KÃ¤hler-Einstein equation is through various continuity methods. Traditionally, one uses Aubin's continuity method involving twisted KE metrics. A more recent continuity method uses KE metrics with edge cone singularities as proposed by Donaldson. Finally, there is a KÃ¤hler-Ricci flow approach. Tian's early works showed that the most difficult part in proving the YTD conjecture by continuity methods is to establish the algebraicity of limit objects in the Gromov-Hausdorff topology, and he had essentially reduced this difficulty to proving some partial

C^{0}

-estimates. The partial

C^{0}

estimates were later proved in different settings, starting with Donaldson-Sun's work in the KÃ¤hler-Einstein case, which led to the solution of the YTD conjecture for smooth Fano manifolds in [24,65]. Moreover, the partial

C^{0}

-estimates has applications in constructing moduli spaces of smoothable KÃ¤hler-Einstein varieties and proving quasiprojectivity of the moduli spaces of KE manifolds, and these applications preclude the algebraic approach mentioned earlier (see [68]). We also refer to [31,66] for surveys on related topics in this approach.

Very recently, yet another quantization approach is carried out by Kewei Zhang based partly on an earlier work of Rubinstein-Tian-Zhang. Zhang considered an analytic invariant of Moser-Trudinger type, namely

δ^{A} (X) = sup {c : sup_{φ \in H} \int_{X} e^{- c (φ - E (φ))} < + \infty}

It is easy to show that the coercivity of

D

-functional is equivalent to

δ^{A} (X) > 1

. The authors of [61] introduced a quantization

δ_{m}^{A} (X)

by using a quantization of

E

on the space of Bergman metrics, and further proved

δ_{m}^{A} (X) = δ_{m} (X)

. Using some deep results in complex geometry including Tian's work on Bergman kernels and Berndtsson's subharmonicity theorem, it is proved in [73] that

lim_{m \to + \infty} δ_{m}^{A} (X) = δ^{A} (X)

. Combining these discussions with the algebraic convergence result of Blum-Jonsson and the valuative criterion of uniform stability of Fujita discussed earlier, Zhang gets

δ^{A} (X) = δ (X)

and completes the proof of uniform version of YTD conjecture for smooth Fano manifolds. It would be interesting to extend this approach to the more general case (i.e., Fano varieties with continuous automorphism groups).

We finish by remarking that it is of interest to apply the ideas and methods from the above two approaches to study the YTD conjecture for general polarizations. For the approach involving partial

C^{0}

-estimates, the geometry is complicated by collapsing phenomenon in the Gromov-Hausdorff convergence with only scalar curvature bounds, which is very difficult to study with current techniques. For the quantization approach, there were some attempts by Mabuchi in several works. But the precise picture seems again unclear.

ACKNOWLEDGMENTS

The author thanks Gang Tian, Chenyang Xu, Feng Wang, Xiaowei Wang, Jiyuan Han, and Yuchen Liu for collaborations on works discussed in this survey. He also thanks many mathematicians for various discussions related to our works: Vestislav Apostolov, Robert Berman, SÃ©bastien Boucksom, TamÃ¡s Darvas, Simon Donaldson, Kento Fujita, HansJoachim Hein, Mattias Jonsson, LÃ¡szlÃ³ Lempert, Mircea MustaÅ£Äƒ, Yuji Odaka, Yanir Rubinstein, Jian Song, Song Sun, Jun Yu, and many others. A large part of the author's works discussed here were carried out at the Department of Mathematics, Purdue University, and he is grateful to his colleagues and friends there for creating a great environment of research.

FUNDING

This work is partially supported by NSF grant (DMS-181086) and an Alfred Sloan Fellowship.

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CHI LI

Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd. Piscataway, NJ 08854-8019, USA, chi.li @ rutgers.edu

THE DOUBLE RAMIFICATION CYCLE FORMULA

AARON PIXTON

ABSTRACT

The double ramification cycle

{D R}_{g} (A) = {D R}_{g} (μ, ν)

is a cycle in the moduli space of stable curves parametrizing genus

g

curves admitting a map to

P^{1}

with specified ramification profiles

μ, ν

over two points. In 2016, Janda, Pandharipande, Zvonkine, and the author proved a formula expressing the double ramification cycle in terms of basic tautological classes, answering a question of Eliashberg from 2001. This formula has an intricate combinatorial shape involving an unusual way to sum divergent series using polynomial interpolation. Here we give some motivation for where this formula came from, relating it both to an older partial formula of Hain and to Givental's R-matrix action on cohomological field theories.

MATHEMATICS SUBJECT CLASSIFICATION 2020

Primary 14N35; Secondary 14H10, 14C17

KEYWORDS

Moduli of curves, double ramification cycle

1. INTRODUCTION

Let

g, n

be nonnegative integers satisfying

2 g - 2 + n > 0

, so that the moduli space

{\bar{M}}_{g, n}

of stable curves of genus

g

with

n

markings is nonempty. Let

A = (a_{1}, \dots, a_{n}) \in Z^{n}

be a vector of

n

integers satisfying

a_{1} + \dots + a_{n} = 0

. In this paper we will be interested in a Chow cycle class

{DR}_{g} (A) \in A^{g} ({\bar{M}}_{g, n})

that depends on this data.

There are two main perspectives on how to think about and define

{DR}_{g} (A)

, the double ramification cycle. The first is the source of its name; we think of it as parametrizing the genus

g

curves

C

that admit a finite map

C \to P^{1}

with specified ramification profiles

μ, ν

over two points (say 0 and

\infty

). These two ramification profiles are encoded in the vector

A

: we can take the positive and negative components of

A

separately to get two partitions of equal size. The marked points with nonzero

a_{i}

should then be the points in

C

lying above 0 and

\infty

, while the marked points with

a_{i} = 0

are unconstrained. Ramification above points other than 0 and

\infty

is unconstrained.

The above description defines a double ramification locus inside the moduli space of smooth curves

M_{g, n}

that is usually (but not always) of pure codimension

g

. To extend this to a codimension

g

class on

{\bar{M}}_{g, n}

, we can use the virtual class in relative GromovWitten theory. There is a moduli space of stable (rubber) maps to

P^{1}

with given marked ramification over two points,

{\bar{M}}_{g, n} {(P^{1} / {0, \infty}, μ, ν)}^{\sim}

, equipped with a forgetful map

p

{\bar{M}}_{g, n} {(P^{1} / {0, \infty}, μ, ν)}^{\sim} \to {\bar{M}}_{g, n}

, and the double ramification cycle can be taken to be the pushforward under this map of the virtual class,

{DR}_{g} (A) = p_{*} {[{\bar{M}}_{g, n} {(P^{1} / {0, \infty}, μ, v)}^{\sim}]}^{v i r} \in A^{g} ({\bar{M}}_{g, n})

The second perspective on

{DR}_{g} (A)

is via Abel-Jacobi maps. Let

X_{g} \to A_{g}

be the universal abelian variety of dimension

g

. Then the data in the vector

A

can be used to define a morphism

j_{A} : M_{g, n} \to X_{g}

(C, p_{1}, \dots, p_{n}) \mapsto (Jac (C), O_{C} (a_{1} p_{1} + \dots + a_{n} p_{n}))

The double ramification locus is then the inverse image under this map of the zero section

Z_{g}

X_{g}

, since

C

admits a map to

P^{1}

with the given ramification profiles if and only if

O_{C} (a_{1} p_{1} + \dots + a_{n} p_{n})

is trivial.

This Abel-Jacobi map extends easily to

M_{g, n}^{c t}

, the moduli space of curves of compact type (those with compact Jacobians), but using this perspective to define the double ramification cycle on all of

{\bar{M}}_{g, n}

requires more work. It also is not obvious that constructing

{DR}_{g} (A)

in this way will give the same class as that given by relative Gromov-Witten theory, even after restriction to

M_{g, n}^{c t}

. For one approach to these questions using logarithmic and tropical geometry, see the work of Marcus and Wise [13].

Eliashberg proposed the problem of giving a formula for the double ramification cycle in 2001, in the context of symplectic field theory. This problem was solved by Janda, Pandharipande, Zvonkine, and the author in 2016 [11], giving an explicit combinatorial formula for the double ramification cycle. This formula has an unexpected form-an additional
integer parameter

r > 0

is introduced, then an expression is written down that becomes polynomial in

r

for

r

sufficiently large, and finally this polynomial is specialized to

r = 0

. Subsequent papers extending or generalizing the double ramification cycle formula in various ways (e.g.,

[2, 5, 12]

) have left the basic combinatorial structure of the formula virtually unchanged. The purpose of this paper is to discuss this structure and give some motivation for where it comes from.

In Section 2, we review the tautological classes in the Chow ring of the moduli space of stable curves. In Section 3, we discuss results leading up to the formula of [11], most notably Hain's formula for the compact type double ramification cycle. Section 4 is the heart of the paper and consists of an extended discussion motivating the shape of the double ramification cycle formula. We conclude in Section 5 by stating the formula and briefly explaining how its proof in [11] is related to some of the motivation in Section 4.

2. TAUTOLOGICAL CLASSES

2.1. Preliminaries

In this section we review the language in which the double ramification cycle formula is written. This is the language of the tautological ring, a subring

R^{*} ({\bar{M}}_{g, n}) \subseteq A^{*} ({\bar{M}}_{g, n})

containing most classes that arise naturally in geometry.

Following Faber and Pandharipande [6], the tautological rings

R^{*} ({\bar{M}}_{g, n})

can be defined simultaneously for all

g, n \geq 0

satisfying

2 g - 2 + n > 0

as the smallest subrings of the Chow rings

A^{*} ({\bar{M}}_{g, n})

closed under pushforward by forgetful maps

{\bar{M}}_{g, n + 1} \to {\bar{M}}_{g, n}

and gluing maps

{\bar{M}}_{g, n + 2} \to {\bar{M}}_{g + 1, n}

{\bar{M}}_{g_{1}, n_{1} + 1} \times {\bar{M}}_{g_{2}, n_{2} + 1} \to {\bar{M}}_{g_{1} + g_{2}, n_{1} + n_{2}}

. Our discussions of tautological classes will use a more explicit description of them. Graber and Pandharipande [8, APPENDIX A] gave a set of additive generators and a multiplication law satisfied by these generators.

These additive generators are formed from three ingredients: psi classes, kappa classes, and generalized gluing maps corresponding to stable graphs. The psi classes

ψ_{i} \in A^{1} ({\bar{M}}_{g, n}), i = 1, \dots, n

correspond to the

n

marked points and are defined as the first Chern classes of the cotangent line bundles to the curves at those points. The ArbarelloCornalba [1] kappa classes are then the pushforwards of powers of psi classes,

κ_{a} := π_{*} (ψ_{n + 1}^{a + 1}) \in A^{a} ({\bar{M}}_{g, n})

where

π : {\bar{M}}_{g, n + 1} \to {\bar{M}}_{g, n}

forgets the last marking. The kappa classes will not appear in any of the formulas in this paper.

The tautological ring of the moduli space of smooth marked curves,

R^{*} (M_{g, n})

, is the ring generated by these

ψ_{i}

and

κ_{a}

. To extend this to

R^{*} ({\bar{M}}_{g, n})

we need classes supported on boundary strata.

2.2. Stable graphs

A stable graph

Γ

is the combinatorial data of a boundary stratum in

{\bar{M}}_{g, n}

. It consists of the following:

(1) a set of vertices

V (Γ)

;

(2) a genus

g_{v} \geq 0

at each vertex

v \in V (Γ)

;

(3) a set of half-edges

H (Γ)

;

(4) an incidence map

H (Γ) \to V (Γ)

;

(5) a partition of

H (Γ)

into sets of size 2 (called edges, the set of which is denoted

E (Γ))

and sets of size 1 (called legs);

(6) a bijection between the set of legs and

{1, \dots, n}

The underlying graph is required to be connected. The stability constraint is that

2 g_{v} - 2 + n_{v} > 0

at each vertex

v

, where

n_{v}

is the number of half-edges incident to

v

. The genera are constrained by the identity

2 g - 2 + n = \sum_{v \in V (Γ)} (2 g_{v} - 2 + n_{v})

or equivalently that

g - \sum_{v} g_{v} = h^{1} (Γ)

, the first Betti number of the graph. Such a stable graph

Γ

corresponds to a generalized gluing map

ξ_{Γ} : \prod_{v \in V (Γ)} {\bar{M}}_{g_{v}, n_{v}} \to {\bar{M}}_{g, n}

We can then consider classes

ξ_{Γ *} (α) \in A^{*} ({\bar{M}}_{g, n})

where

Γ

is a stable graph and

α

is a monomial in the psi and kappa classes on the

{\bar{M}}_{g_{v}, n_{v}}

factors. These are the additive generators for the tautological ring considered in [8].

2.3. Compact type

The moduli space of curves of compact type, denoted

M_{g, n}^{c t}

, is the open subscheme of

{\bar{M}}_{g, n}

consisting of those curves whose dual graph is a tree. Its tautological ring

R^{*} (M_{g, n}^{c t})

is the image of

R^{*} ({\bar{M}}_{g, n})

under restriction, so it is additively generated by classes

ξ_{Γ_{*}} (α)

as above where

Γ

is a tree.

It will be convenient for us to have notation for the compact type boundary divisor classes when stating Hain's formula below, (3.2). If

Γ

is a stable graph with 2 vertices and 1 edge and one of the vertices is genus

h

and has those legs with markings in a set

S \subseteq 1, 2, \dots, n

, let

δ_{h, S} = ξ_{Γ_{*}} (1)

be the corresponding boundary divisor class.

3. PREVIOUS FORMULAS AND RESULTS

The first progress towards a formula for the double ramification cycle was when Faber and Pandharipande [7] proved that the double ramification cycle lies in the tautological ring, and thus in theory must be expressible in terms of the generators described in
the previous section. Their proof, although in principle constructive, involves a complicated recursion and does not seem to yield a practical formula.

The first progress towards an explicit formula came when Hain [10] computed the double ramification cycle when restricted to the compact type locus

M_{g, n}^{c t}

. On this locus the double ramification cycle is the pullback along an Abel-Jacobi map

j_{A} : M_{g, n}^{c t} \to X_{g, n}

of the class of the zero section

Z_{g, n}

of the universal abelian variety

X_{g, n} \to A_{g, n}

. Hain showed that the class of this zero section is

[Z_{g, n}] = Θ^{g} / g

! and computed the pullback of the theta divisor

Θ

as an explicit divisor on

M_{g, n}^{c t}

\begin{matrix} (3.1) & j_{A}^{*} Θ = \sum_{i = 1}^{n} \frac{a_{i}^{2}}{2} ψ_{i} - \sum_{h, S} \frac{a_{S}^{2}}{4} δ_{h, S} \end{matrix}

where

a_{S} = \sum_{i \in S} a_{i}

and the second sum runs over all

h, S

defining boundary divisor classes.

Hain's formula for the compact type double ramification cycle is then

{DR}_{g}^{c t} (A) = \frac{1}{g!} {(j_{A}^{*} Θ)}^{g} = \frac{1}{g!} {(\sum_{i = 1}^{n} \frac{a_{i}^{2}}{2} ψ_{i} - \sum_{h, S} \frac{a_{S}^{2}}{4} δ_{h, S})}^{g}

The divisor formula (3.1) is a homogeneous polynomial of degree 2 in

A

, so Hain's DR formula (3.2) is a homogeneous polynomial of degree

2 g

A

Grushevsky and Zakharov [9] extended Hain's computation slightly, expanding from

M_{g, n}^{c t}

to a slightly larger open subscheme of

{\bar{M}}_{g, n}

by adding the locus of curves whose dual graph is a tree with a single loop added at one vertex. If

Γ

is the stable graph with a single vertex and single loop, then their correction term is the codimension

g

part of

\begin{matrix} (3.3) & ξ_{Γ *} (- \prod_{i = 1}^{n} \exp (\frac{1}{2} a_{i}^{2} ψ_{i}) \sum_{k = 1}^{\infty} \frac{B_{2 k}}{2^{k} k!} {(ψ + ψ^{'})}^{k - 1}) \end{matrix}

where

ψ_{1}, \dots, ψ_{n}

are the psi classes on the legs,

ψ, ψ^{'}

are the psi classes on the two halfedges of the loop, and

B_{2 k}

is a Bernoulli number.

In particular, the double ramification cycle is no longer a homogeneous polynomial in

A

when computed beyond compact type. This was also seen in work of Buryak, Shadrin, Spitz, and Zvonkine [3], who showed that the top degree intersections of double ramification cycles with monomials in the psi classes are inhomogeneous polynomials of degree

2 g

A

4. MOTIVATION FOR THE FORMULA

In this section we discuss various observations and ideas that come about when

one tries to extend Hain's formula (3.2) to

{\bar{M}}_{g, n}

to obtain a full double ramification cycle formula.

4.1. Expanding Hain's formula

Exponentiating a boundary divisor class can be done using the multiplication laws for tautological classes [8, APPENDIX A]. Multiplying out Hain's formula (3.2) in this way gives
a nice sum over trees:

{DR}_{g}^{c t} (A)

is the codimension

g

part of

\begin{aligned} \sum_{T stable tree} \frac{1}{| Aut (T) |} {(ξ_{T})}_{*} [\prod_{i = 1}^{n} \exp (\frac{1}{2} a_{i}^{2} ψ_{h_{i}}) \\ (1.1) & \cdot \prod_{e = {h, h^{'}} \in E (T)} \frac{1 - \exp (- \frac{1}{2} w (h) w (h^{'}) (ψ_{h} + ψ_{h^{'}}))}{ψ_{h} + ψ_{h^{'}}}] \end{aligned}

where the function

w : H (T) \to Z

is defined here by contracting all the edges in the tree

T

other than the one containing

h

and then letting

w (h)

be the sum of the

a_{i}

for the legs

i

on the same vertex as the half-edge

h

Extending this formula to

{\bar{M}}_{g, n}

requires us to provide a polynomial (or power series) in the psi classes for every stable graph

Γ

, not just every stable tree. The

w (h)

definition above does not naturally extend to non-separating edges, so it is not immediately clear how to do this. Moreover, we know that this power series needs to be (3.3) for the single-loop graph, so something quite new is going on even there.

4.2. Cohomological field theory axioms

A cohomological field theory (CohFT) is a collection of classes

Ω_{g, n} (γ_{1}, \dots, γ_{n})

{\bar{M}}_{g, n}

for all

g

and

n

, where the inputs

γ_{i}

belong to some finite set

S

(a basis for the state space of the CohFT). These classes must satisfy certain compatibility axioms relating them to each other under pullback by natural maps between the

{\bar{M}}_{g, n}

. For one basic treatment of CohFTs and Givental's R-matrix action, see [14]. The double ramification cycle is not quite a CohFT, but it satisfies some subset of the properties of one. For example, if

j : {\bar{M}}_{g_{1}, n_{1} + 1} \times {\bar{M}}_{g_{2}, n_{2} + 1} \to {\bar{M}}_{g, n}

is a separating gluing map where the marked points split into sets

I_{1}, I_{2}

with

| I_{i} | = n_{i}

, then we have

j^{*} {DR}_{g} (a_{1}, \dots, a_{n}) = {DR}_{g_{1}} ({a_{i} ∣ i \in I_{1}}, t) \otimes {DR}_{g_{2}} ({a_{i} ∣ i \in I_{2}}, - t)

where

t \in Z

is the unique insertion that makes the parameters sum to 0 in each DR term on the right.

If the double ramification cycle were a CohFT, we would want a similar formula for the pullback along the nonseparating gluing map

k : {\bar{M}}_{g - 1, n + 2} \to {\bar{M}}_{g, n}

: the natural thing to write down would be

k^{*} {DR}_{g} (a_{1}, \dots, a_{n}) = \sum_{t \in Z} {DR}_{g - 1} (a_{1}, \dots, a_{n}, t, - t)

but it is not clear how one might make sense of this infinite sum-it will not converge in any standard sense. What is going wrong here is that CohFTs are supposed to depend multilinearly on parameters from a finite-dimensional state space, but double ramification cycles take inputs in

Z

so the state space appears to be infinite-dimensional.

So the double ramification cycle behaves like a CohFT as far as separating nodes are concerned, but the state space would have to be infinite-dimensional and this makes it unclear what to do at nonseparating nodes.

4.3. Givental's R-matrix action

Teleman [16] proved that semisimple CohFTs all have a very particular graph sum form, given by applying Givental's R-matrix action to a CohFT that lives fully in codimension zero. The rough shape of the resulting formula for a semisimple CohFT is

\begin{aligned} Ω_{g, n} (γ_{1}, \dots, γ_{n}) = & \sum_{Γ stable graph} \sum_{w : H (Γ) \to S} \frac{1}{| Aut (Γ) |} {(ξ_{Γ})}_{*} [ \\ \prod_{v \in V (Γ)} (vertex factor) \prod_{i = 1}^{n} (leg factor) \prod_{e = {h, h^{'}} \in E (Γ)} (edge factor)] \end{aligned}

where the second sum is over functions

w

on the half-edges of the graph taking values in some set

S

(a basis for the state space of the CohFT) and the values of

w

on the legs

h_{1}, \dots, h_{n}

are given as

w (h_{i}) = γ_{i}

. The various factors are then power series (that depend on

w

) in the corresponding kappa and psi classes. The expanded version of Hain's compact type formula (4.1) is of this shape: we take

S = Z

, the vertex factor is 0 unless all of the incident

w (h)

sum to zero, and the edge factor is 0 unless the two

w (h)

along the edge sum to zero. These vanishings effectively place the following constraints on

w

(to get a nonzero contribution to

{D R}_{g}^{c t} (A))

(1)

w (h_{i}) = a_{i}

for

i = 1, 2, \dots, n

, where

h_{i}

is the

i

th leg;

(2)

w (h) + w (h^{'}) = 0

{h, h^{'}}

is an edge;

(3)

\sum_{h \to v} w (h) = 0

for each vertex

v

We say

w

is balanced (with respect to

A

) if it satisfies these constraints. In other words,

w

is a flow on

Γ

with sources/sinks at the legs (with specified values given there by

A

). When

Γ

is a tree, there is a unique such balanced

w

and we recover the

w (h)

used in (4.1).

From this perspective it is natural to just try to take (4.1) and extend it to be a Givental-type sum over arbitrary graphs (not just trees), but then there will be infinitely many choices of

w

and the resulting infinite sums will be nonconvergent. Moreover, careful comparison with the exact form of Givental's R-matrix action suggests that the vertex factor should contribute a total factor of something like "

| Z |^{- h^{1} (Γ)}

." Note that the set of balanced

w

is a torsor over

H_{1} (Γ; Z) ≅ Z^{h^{1} (Γ)}

, so this factor feels like some sort of infinite averaging procedure.

4.4. Divergent averages

Returning to the simplest non-tree case, the graph with one vertex and one loop, matching things up with (3.3) would then require making sense of the "infinite average" identity

\begin{matrix} (4.2) & \frac{1}{| Z |} \sum_{c \in Z} c^{2 k} = B_{2 k} \end{matrix}

This is reminiscent of the zeta regularization sum

\sum_{c \geq 1} c^{2 k - 1} = ζ (1 - 2 k) = - \frac{B_{2 k}}{2 k}

but there is no obvious way to make sense of this similarity. Moreover, more complicated graphs require much more complicated divergent sums; for example, a graph with two vertices, a double edge between them, and one leg on each vertex gives rise to infinite sums like

\begin{matrix} (4.3) & \frac{1}{| Z |} \sum_{c + d = a} c^{2 k} d^{2 l} \end{matrix}

which must be interpreted.

4.5. Interpolating finite rank CohFTs

The problem with writing down a double ramification cycle formula of this type is clearly that the state space is infinite-dimensional. If we replace

Z

with

Z / r Z

everywhere then there is no difficulty with writing down a similar-looking finite rank CohFT. The result might be something like the following (the case of a diagonal R-matrix-for an example of a more complicated CohFT of this general type, see [15]):

\begin{aligned} \sum_{Γ stable graph} \sum_{\begin{array}{c} w : H (Γ) \to Z / r Z \\ balanced \end{array}} \frac{1}{| Aut (Γ) |} {(ξ_{Γ})}_{*} [\frac{1}{r^{h^{1} (Γ)}} \prod_{i = 1}^{n} \exp (F_{w (h_{i})} (ψ_{h_{i}})) \\ \cdot \prod_{e = {h, h^{'}} \in E (Γ)} \frac{1 - \exp (F_{w (h)} (ψ_{h}) + F_{w (h^{'})} (ψ_{h^{'}}))}{ψ_{h} + ψ_{h^{'}}}], \end{aligned}

for power series

F_{a} (Z)

for

a \in Z / r Z

with

F_{0} (Z) = 0

and

F_{- a} (- Z) = - F_{a} (Z)

If we take

F_{a} (Z) = \frac{1}{2} a^{2} Z

for

- \frac{r}{2} < a \leq \frac{r}{2}

then this CohFT starts to look very much like the expanded version of Hain's formula, (4.1). In fact, if

Γ

is a tree then the

Γ

-term in this sum agrees with that in Hain's formula for all sufficiently large

r

. So it is tempting to try to take the limit as

r \to \infty

of these CohFTs. But this is not quite right: the

r

-version of the left side of (4.2) is then

\frac{1}{r} \sum_{- \frac{r}{2} < c \leq \frac{r}{2}} c^{2 k}

This certainly does not converge as

r \to \infty

. However, if we restrict to even

r

then it is polynomial in

r

, and if we examine the coefficients of this polynomial then we see that

B_{2 k}

, the desired value, is the constant coefficient in

r

This suggests a potential interpretation even of more complicated sums like (4.3):

\begin{aligned} \frac{1}{| Z |} \sum_{c + d = a} c^{2 k} d^{2 l} & = \frac{1}{| Z / 0 Z |} \sum_{\begin{array}{c} c, d \in Z / 0 Z \\ c + d = a (mod 0) \end{array}} c^{2 k} d^{2 l} \\ := {[\frac{1}{| Z / r Z |} \sum_{\begin{array}{c} c, d \in Z / r Z \\ c + d = a (mod r) \end{array}} c^{2 k} d^{2 l}]}_{r = 0} \end{aligned}

where

c

and

d

must be interpreted inside

c^{2 k} d^{2 l}

as elements of

Z

via some choice of

mod r

representatives (we used

- r / 2 + 1, \dots, r / 2

before but

0, \dots, r - 1

will give the same final answer) and setting

r = 0

at the end is done by polynomial interpolation.

4.6. Geometric interpretation from $(k / r)$ -spin structures

(k / r)

-spin structure on a smooth curve

C

with marked points

p_{i}

and weights

a_{i}

is a choice of line bundle

L

C

such that

L^{\otimes r} \equiv ω_{C}^{\otimes k} (a_{1} p_{1} + \dots + a_{n} p_{n})

. If we take

k = 0

and assume the weights

a_{i}

sum to zero then for any positive

r

any smooth curve will have such "

r

th root structures." But we can also interpret this construction as meaningful when

r = 0

, when we get that a curve only admits a (0/0)-spin structure if it is in the double ramification locus. This observation gives a vague geometric idea for what it might mean to think of the double ramification cycle as given by specializing some parameter

r

to 0 .

5. THE DOUBLE RAMIFICATION CYCLE FORMULA

We can now state the main result of [11], the double ramification cycle formula:

Theorem 1 ([11]).

{D R}_{g} (A)

is the codimension

g

part of

\begin{aligned} \sum_{Γ stable graph} \frac{1}{| Aut (Γ) |} {(ξ_{Γ})}_{*} [\frac{1}{| Z |^{h^{1} (Γ)}} \sum_{\begin{array}{c} w : H (Γ) \to Z \\ balanced \end{array}} \prod_{i = 1}^{n} \exp (\frac{1}{2} a_{i}^{2} ψ_{h_{i}}) \\ \cdot \prod_{e = {h, h^{'}} \in E (Γ)} \frac{1 - \exp (- \frac{1}{2} w (h) w (h^{'}) (ψ_{h} + ψ_{h^{'}}))}{ψ_{h} + ψ_{h^{'}}}] \end{aligned}

where formal expressions of the form

\frac{1}{| Z |^{h^{1} (Γ)}} \sum_{\begin{matrix} w : H (Γ) \to Z \\ balanced \end{matrix}} P ({w (h)})

(for

P

a polynomial) are evaluated by setting

r = 0

in the corresponding

r

-polynomial

\frac{1}{r^{h^{1} (Γ)}} \sum_{\begin{matrix} w : H (Γ) \to {0, 1, \dots, r - 1} \\ balanced mod r \end{matrix}} P ({w (h)})

The combinatorial result (necessary for this theorem statement to make sense) that the expression in the final line is in fact a polynomial in

r

(for

r

sufficiently large) was proved in [11, APPENDIX A].

The proof of Theorem 1 in [11] follows some of the motivation in Section 4. We first explain the meaning of the additional

r

parameter. For each

r > 0

, let

P^{1} [r]

denote the projective line with a

B Z_{r}

orbifold point at 0 . One can then use

C^{*}

-localization on the moduli space of relative stable maps to

P^{1} [r] / {\infty}

to obtain complicated relations that entangle double ramification cycles, classes coming from moduli of

(0 / r)

-spin curves (discussed briefly in the case of smooth curves in Section 4.6), and other basic tautological classes. The relevant

(0 / r)

-spin classes were previously computed by Chiodo [4] using GrothendieckRiemann-Roch.

These localization relations are too difficult to study effectively for specific values of

r

, but it turns out that they have polynomial dependence on

r

. Taking the constant term in

r

simplifies them greatly: most of the terms vanish, and the only remaining terms are a
single double ramification cycle and the

r = 0

interpolation of certain classes written in terms of the Chern characters of the pushforward of the universal

r

th root line bundle on the moduli space of

(0 / r)

-spin curves. Chiodo's formula [4] for these Chern characters gives that these classes are CohFTs with formulas of the type described in Section 4.5. The power series in psi classes appearing in these formulas do not look exactly like those appearing in Theorem 1, but they have the same

r = 0

interpolation. (In the language of Section 4.5, the power series

F_{a} (Z)

will be congruent to

\frac{1}{2} a^{2} Z mod r

.) The result is a proof of Theorem 1 .

ACKNOWLEDGMENTS

The author is grateful to R. Cavalieri, D. Holmes, F. Janda, S. Grushevsky, S. Molcho, G. Oberdieck, R. Pandharipande, J. Schmitt, D. Zakharov, and D. Zvonkine for many discussions about double ramification cycles and related topics over the years.

FUNDING

The author was partially supported by National Science Foundation Grant No. 1807079.

REFERENCES

[1] E. Arbarello and M. Cornalba, Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves. J. Algebraic Geom. 5 (1996), no. 4, 705-749.

[2] Y. Bae, D. Holmes, R. Pandharipande, J. Schmitt, and R. Schwarz, Pixton's formula and Abel-Jacobi theory on the Picard stack. 2021, arXiv:2004.08676v2.

[3] A. Buryak, S. Shadrin, L. Spitz, and D. Zvonkine, Integrals of

ψ

-classes over double ramification cycles. Amer. J. Math. 137 (2015), no. 3, 699-737.

[4] A. Chiodo, Towards an enumerative geometry of the moduli space of twisted curves and

r

th roots. Compos. Math. 144 (2008), no. 6, 1461-1496.

[5] E. Clader and F. Janda, Pixton's double ramification cycle relations. Geom. Topol. 22 (2018), no. 2, 1069-1108.

[6] C. Faber and R. Pandharipande, Logarithmic series and Hodge integrals in the tautological ring. Michigan Math. J. 48 (2000), 215-252.

[7] C. Faber and R. Pandharipande, Relative maps and tautological classes. J. Eur. Math. Soc. (JEMS) 7 (2005), no. 1, 13-49.

[8] T. Graber and R. Pandharipande, Constructions of nontautological classes on moduli spaces of curves. Michigan Math. J. 51 (2003), no. 1, 93-109.

[9] S. Grushevsky and D. Zakharov, The zero section of the universal semiabelian variety and the double ramification cycle. Duke Math. J. 163 (2014), no. 5, 953-982.

[10] R. Hain, Normal functions and the geometry of moduli spaces of curves. In Handbook of moduli. Vol. I, pp. 527-578, Adv. Lect. Math. (ALM) 24, Int. Press, Somerville, MA, 2013.

[11] F. Janda, R. Pandharipande, A. Pixton, and D. Zvonkine, Double ramification cycles on the moduli spaces of curves. Publ. Math. Inst. Hautes Ã‰tudes Sci. 125 (2017), no. 1, 221-266.

[12] F. Janda, R. Pandharipande, A. Pixton, and D. Zvonkine, Double ramification cycles with target varieties. J. Topol. 13 (2020), no. 4, 1725-1766.

[13] S. Marcus and J. Wise, Logarithmic compactification of the Abel-Jacobi section. Proc. Lond. Math. Soc. (3) 121 (2020), no. 5, 1207-1250.

[14] R. Pandharipande, A. Pixton, and D. Zvonkine, Relations on

{\bar{M}}_{g, n}

via 3-spin structures. J. Amer. Math. Soc. 28 (2015), no. 1, 279-309.

[15] R. Pandharipande, A. Pixton, and D. Zvonkine, Tautological relations via

r

-spin structures. J. Algebraic Geom. 28 (2019), no. 3, 439-496.

[16] C. Teleman, The structure of 2D semi-simple field theories. Invent. Math. 188 (2012), no. 3, 525-588.

AARON PIXTON

Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA, pixton@umich.edu

EFFECTIVE RESULTS IN THE THREE-DIMENSIONAL MINIMAL MODEL PROGRAM
YURI PROKHOROV

ABSTRACT

We give a brief review on recent developments in the three-dimensional minimal model program.

MATHEMATICS SUBJECT CLASSIFICATION 2020

Primary 14E30; Secondary 14J30, 14B05, 14E05

KEYWORDS

Minimal model, Mori contraction, terminal (canonical) singularity, flip, extremal ray

In this note we give a brief review on recent developments in the three-dimensional minimal model program (MMP for short). Certainly, this is not a complete survey of all advances in this area. For example, we do not discuss the minimal models of varieties of nonnegative Kodaira dimension, as well as applications to birational geometry and moduli spaces.

The aim of the MMP is to find a good representative in a fixed birational equivalence class of algebraic varieties. Starting with an arbitrary smooth projective variety, one can perform a finite number of certain elementary transformations, called divisorial contractions and flips, and at the end obtain a variety which is simpler in some sense. Most parts of the MMP are completed in arbitrary dimension. One of the basic remaining problems is the following:

Describe all the intermediate steps and the outcome of the MMP.

The MMP makes sense only in dimensions

\geq 2

, and for surfaces it is classical and well known. So the first nontrivial case is the three-dimensional one. It turns out that to proceed with the MMP in dimension

\geq 3

, one has to work with varieties admitting certain types of very mild, the so-called terminal, singularities. On the other hand, dimension 3 is the last dimension where one can expect effective results: in higher dimensions, classification results become very complicated and unreasonably long.

We will work over the field

C

of complex numbers throughout. A variety is either an algebraic variety or a reduced complex space.

1. SINGULARITIES

Recall that a Weil divisor

D

on a normal variety is said to be

Q

-Cartier if its multiple

n D

, for some

n

, is a Cartier divisor. For any morphism

f : Y \to X

, the pull-back

f^{*} D

of a

Q

-Cartier divisor

D

is well defined as a divisor with rational coefficients (

Q

-divisor). A variety

X

has

Q

-factorial singularities if any Weil divisor on

X

Q

-Cartier.

Definition 1.1. A normal algebraic variety (or an analytic space)

X

is said to have terminal (resp. canonical, log terminal, log canonical) singularities if the canonical Weil divisor

K_{X}

Q

-Cartier and, for any birational morphism

f : Y \to X

, one can write

\begin{matrix} (1.1.1) & K_{Y} = f^{*} K_{X} + \sum a_{i} E_{i} \end{matrix}

where

E_{i}

are all the exceptional divisors and

a_{i} > 0

(resp.

a_{i} \geq 0, a_{i} > - 1, a_{i} \geq - 1

) for all

i

. The smallest positive

m

such that

m K_{X}

is Cartier is called the Gorenstein index of

X

. Canonical singularities of index 1 are rational Gorenstein.

The class of terminal

Q

-factorial singularities is the smallest class that is closed under the MMP. Canonical singularities are important because they appear in the canonical models of varieties of general type. A crucial observation is that terminal singularities lie in codimension

\geq 3

. In particular, terminal surface singularities are smooth and terminal threefold singularities are isolated. Canonical singularities of surfaces are called

D u V a l

or rational double points. Any two-dimensional log terminal singularity is a quotient of a smooth surface germ by a finite group [32]. Terminal threefolds singularities were classified by M. Reid [65] and S. Mori [43].

Example. Let

X \subset C^{4}

be a hypersurface given by the equation

ϕ (x_{1}, x_{2}, x_{3}) + x_{4} ψ (x_{1}, \dots, x_{4}) = 0

where

ϕ = 0

is an equation of a

D u Val

(ADE) singularity. Then the singularity of

X

at 0 is canonical Gorenstein. It is terminal if and only if it is isolated. Singularities of this type are called cDV.

According to [65], any three-dimensional terminal singularity of index

m > 1

is a quotient of an isolated cDV-singularity by the cyclic group

μ_{m}

of order

m

. More precisely, we have the following

Theorem 1.2 ([65]). Let

(X ∋ P)

be an analytic germ of a three-dimensional terminal singularity of index

m \geq 1

. Then there exist an isolated

c D V

-singularity

(X^{♯} ∋ P^{♯})

and a cyclic

μ_{m}

-cover

π : (X^{♯} ∋ P^{♯}) ⟶ (X ∋ P)

which is Ã©tale outside

P

The morphism

π

in the above theorem is called the index-one cover. A detailed classification of all possibilities for the equations of

X^{♯} \subset C^{4}

and the actions of

μ_{m}

was obtained in [43] (see also [66]).

Example. Let the cyclic group

μ_{m}

act on

C^{n}

diagonally via

(x_{1}, \dots, x_{n}) \mapsto (ζ^{a_{1}} x_{1}, \dots, ζ^{a_{n}} x_{n}), ζ = ζ_{m} = \exp (2 π i / m)

Then we say that

(a_{1}, \dots, a_{n})

is the collection of weights of the action. Assume that the action is free in codimension 1 . Then the quotient singularity

C^{n} / μ_{m} ∋ 0

is said to be of type

\frac{1}{m} (a_{1}, \dots, a_{n})

. According to the criterion (see [66, THEOREM 4.11]), this singularity is terminal if and only if

\sum_{i = 1}^{n} \bar{k a_{i}} > m for k = 1, \dots, m - 1

where - is the smallest residue mod

m

. In the threefold case this criterion has a very simple form: a quotient singularity

C^{m} / μ_{m}

is terminal if and only if it is of type

\frac{1}{m} (1, - 1, a)

, where

\gcd (m, a) = 1

. This is a cyclic quotient terminal singularity.

Example (

[43, 66]

). Let the cyclic group

μ_{m}

act on

C^{4}

diagonally with weights

(1, - 1, a, 0)

, where

\gcd (m, a) = 1

. Then for a polynomial

ϕ (u, v)

, the singularity at 0 of the quotient

{x_{1} x_{2} + ϕ (x_{3}^{m}, x_{4}) = 0} / μ_{m}

is terminal whenever it is isolated. The index of this singularity equals

m

As a consequence of the classification, we obtain that the local fundamental group of the (analytic) germ of a three-dimensional terminal singularity of index

m

is cyclic of order

m

\begin{matrix} (1.2.1) & π_{1} (X ∖ {P}) ≃ Z / m Z \end{matrix}

In particular, for any Weil

Q

-Cartier divisor

D

X

, its

m

th multiple

m D

is Cartier [32, LEMMA 5.1].

The class of canonical threefold singularities is much larger than the class of terminal ones. However, there are certain boundedness results. For example, it is known that the index of a strictly canonical isolated singularity is at most 6 [31].

The modern higher-dimensional MMP often works with pairs, and one needs to extend Definition 1.1 to a wider class of objects.

Definition. Let

X

be a normal variety and let

B

be an effective

Q

-divisor on

X

. The pair

(X, B)

is said to be

p l t

(resp. lc) if

K_{X} + B

Q

-Cartier and, for any birational morphism

f : Y \to X

, one can write

K_{Y} + B_{Y} = f^{*} (K_{X} + B) + \sum a_{i} E_{i}

where

B_{Y}

is the proper transform of

B, E_{i}

are all the exceptional divisors and

a_{i} > - 1

(resp.

a_{i} \geq - 1)

for all

i

. The pair

(X, B)

is said to be

k l t

if it is plt and

⌊ B ⌋ = 0

2. MINIMAL MODEL PROGRAM

Basic elementary operations in the MMP are Mori contractions.

A contraction is a proper surjective morphism

f : X \to Z

of normal varieties with connected fibers. The exceptional locus of a contraction

f

is the subset

Exc (f) \subset X

of points at which

f

is not an isomorphism. A Mori contraction is a contraction

f : X \to Z

such that the variety

X

has at worst terminal

Q

-factorial singularities, the anticanonical class

- K_{X}

f

-ample, and the relative Picard number

ρ (X / Z)

equals 1. A Mori contraction is said to be divisorial (resp. flipping) if it is birational and the locus

Exc (f)

has codimension 1 (resp.

\geq 2

). For a divisorial contraction, the exceptional locus

Exc (f)

is a prime divisor. A Mori contraction whose target is a lower-dimensional variety is called a Mori fiber space. Then the general fiber is a Fano variety with at worst terminal singularities. In the particular cases where the relative dimension of

X / Z

equals 1 (resp. 2), the Mori fiber space

f : X \to Z

is called a

Q

-conic bundle (resp.

Q

-del Pezzo fibration). If

Z

is a point, then

X

is a Fano variety with at worst terminal

Q

-factorial singularities and

Pic (X) ≃ Z

. For short, we call such varieties

Q

-Fano.

The MMP procedure is a sequence of elementary transformations which are constructed inductively

[35, 39]

. Let

X

be a projective algebraic variety with terminal

Q

-factorial singularities. If the canonical divisor

K_{X}

is not nef, then there exists a Mori contraction

f : X \to Z

. If

f

is divisorial, then

Z

is again a variety with terminal

Q

-factorial singularities and, in this situation, we can proceed with the MMP replacing

X

with

Z

. In contrast,
a flipping contraction takes us out the category of terminal

Q

-factorial varieties. To proceed, one has to perform a surgery operation as follows:

where

f^{+}

is a contraction whose exceptional locus has codimension

\geq 2

and the divisor

K_{X^{+}}

Q

-Cartier and

f^{+}

-ample. Then the variety

X^{+}

again has terminal

Q

-factorial singularities, and we can proceed by replacing

X

with

X^{+}

The process described above should terminate, and at the end we obtain a variety

\bar{X}

such that either

\bar{X}

has a Mori fiber space structure

\bar{X} \to \bar{Z}

K_{\bar{X}}

is nef. One of the remaining open problems is the termination of the program, to be more precise, termination of a sequence of flips. The problem was solved affirmatively in dimension

\leq 4

[35,69], for varieties of general type, for uniruled varieties [5], and in some other special cases. We refer to [3] for more comprehensive survey of the higher-dimensional MMP.

The MMP has a huge number of applications in algebraic geometry. The most impressive consequence of the MMP is the finite generation of the canonical ring

R (X, K_{X}) := ⨁_{n \geq 0} H^{0} (X, O_{X} (m K_{X}))

of a smooth projective variety

X [5, 15]

. Another application of the MMP is the so-called Sarkisov program which allows decomposing birational maps between Mori fiber spaces into composition of elementary transformations, called Sarkisov links [9,16,68]. Also the MMP can be applied to varieties with finite group actions and to varieties over nonclosed fields (see [63]).

As was explained above, the Mori contractions are fundamental building blocks in the MMP. To apply the MMP effectively, one needs to understand the structure of its steps in details. For a Mori contraction

f : X \to Z

of a three-dimensional variety

X

, there are only the following possibilities:

$f$ is divisorial and the image of the (prime) exceptional divisor $E := Exc (f)$ is either a point or an irreducible curve,
$f$ is flipping and the exceptional locus $Exc (f)$ is a union of a finite number of irreducible curves,
$Z$ is a surface and $f$ is a $Q$ -conic bundle,
$Z$ is a curve and $f$ is a $Q$ -del Pezzo fibration,
$Z$ is a point and $X$ is a $Q$ -Fano threefold.

Mori contractions of smooth threefolds to varieties of positive dimension where classified in the pioneering work of S. Mori [42]. S. Cutkosky [12] extended this classification to the case of Gorenstein terminal varieties. Smooth Fano threefolds of Picard number 1 where classified by Iskovskikh [22, 23] (see also [25]). Fano threefolds with Gorenstein terminal singularities are degenerated smooth ones [57]. Below we are going to discuss Mori contractions
of threefolds. We are interested only in the biregular structure of a contraction

f : X \to Z

near a fixed fiber

f^{- 1} (o), o \in Z

. Typically, we do not consider the simple case where

X

is Gorenstein.

3. GENERAL ELEPHANT

A natural way to study higher-dimensional varieties is the inductive one. Typically, to apply this method, we need to find a certain subvariety of dimension one less (divisor) which is sufficiently good is the sense of singularities.

Conjecture 3.1. Let

f : X \to (Z ∋ o)

be a threefold Mori contraction, where

(Z ∋ o)

is a small neighborhood. Then the general member

D \in | - K_{X} |

is a normal surface with Du Val singularities.

The conjecture was proposed by M. Reid who called a good member of

| - K_{X} |

"elephant." We follow this language and call Conjecture 3.1 the General Elephant Conjecture. The importance of the existence of a good member in

| - K_{X} |

is motivated by many reasons:

The general elephant passes through all the non-Gorenstein points of $X$ and so it encodes the information about their types and configuration (cf. Proposition 3.2 below).
For flipping contractions, Conjecture 3.1 is a sufficient condition for the existence of threefold flips [32].
For a divisorial contraction $f : X \to Z$ whose fibers have dimension $\leq 1$ , the image $D_{Z} := f (D)$ of a Du Val elephant $D \in | - K_{X} |$ must be again Du Val and the image $Γ := f (E)$ of the exceptional divisor is a curve on $D_{Z}$ . Then one can reconstruct $f$ starting from the triple $(Z \supset D_{Z} \supset Γ)$ by using a certain birational procedure. Such an approach was successfully worked out in many cases by N. Tziolas [71-74].
If $f : X \to (Z ∋ o)$ is a $Q$ -del Pezzo fibration such that general $D \in | - K_{X} |$ is Du Val, then, composing the projection $D \to Z$ with minimal resolution $\tilde{D} \to D$ , we obtain a relatively minimal elliptic fibration whose singular fibers are classified by Kodaira [36]. Then one can get a bound of multiplicities of fibers and describe the configuration of non-Gorenstein singularities.
For a $Q$ -Fano threefold $X$ , a Du Val general elephant is a (singular) K3 surface. In the case where the linear system $| - K_{X} |$ is "sufficiently big," this implies the existence of a good Gorenstein model [1].

Shokurov [70] generalized Conjecture 3.1 and introduced a new notion which is very efficient in the study of pluri-anticanonical linear systems. Omitting technicalities, we reproduce a weak form of Shokurov's definition.

Definition. An

n

-complement of the canonical class

K_{X}

is a member

D \in | - n K_{X} |

such that the pair

(X, \frac{1}{n} D)

is lc. An

n

-complement is said to be klt (resp. plt) if such is the pair

(X, \frac{1}{n} D)

According to the inversion of adjunction [70], the existence of a Du Val general elephant

D \in | - K_{X} |

is equivalent to the existence of a plt 1-complement. Shokurov developed a powerful theory that works in arbitrary dimension and allows constructing complements inductively (see

[64, 70]

and references therein).

Note that Reid's general elephant fails for Fano threefolds. For example, in

[6, 21]

one can find examples of

Q

-Fano threefolds with an empty anticanonical linear system. Because of this, the statement of Conjecture 3.1 sometimes is called a "principle." Nonetheless, there are only a few examples of such Fano threefolds. In the case

\dim (Z) > 0

, Conjecture 3.1 is expected to be true without exceptions. The following should be considered as the local version of Conjecture 3.1.

Proposition 3.2 (Reid [66]). Let

(X ∋ P

) be the analytic germ of a threefold terminal singularity of index

m > 1

. Then the general member

D \in | - K_{X} |

is a Du Val singularity. Furthermore, let

π : X^{'} \to X

be the index-one cover and let

D^{'} := π^{- 1} (D)

. Then the cover

D^{'} \to D

belongs to one of the following six types:

$(X ∋ P)$	$D^{'} \to D$	$(X ∋ P)$	$D^{'} \to D$
$c A / m$	$A_{k - 1} \overset{m : 1}{\to} A_{k m - 1}$	$c A x / 2$	$A_{2 k - 1} \overset{2 : 1}{\to} D_{k + 2}$
$c A x / 4$	$A_{2 k - 2} \overset{4 : 1}{\to} D_{2 k + 1}$	$c D / 2$	$D_{k + 1} \overset{2 : 1}{\to} D_{2 k}$
$c D / 3$	$D_{4} \overset{3 : 1}{\to} E_{6}$	$c E / 2$	$E_{6} \overset{2 : 1}{\to} E_{7}$

4. DIVISORIAL CONTRACTIONS TO A POINT

In this section we treat divisorial Mori contractions of a divisor to a point. Such contractions are studied very well due to works of Y. Kawamata [34], A. Corti [10], M. Kawakita [26-30], T. Hayakawa [18-20], and others. In this case, General Elephant Conjecture 3.1 has been verified:

Theorem 4.1 (Kawakita [28,29]). Let

f : X \to (Z ∋ o)

be a divisorial Mori contraction that contracts a divisor to a point. Then the general member

D \in | - K_{X} |

is Du Val.

One of the main tools in the proofs is the orbifold Riemann-Roch formula [66]: if

X

is a three-dimensional projective variety with terminal singularities and

D

is a Weil

Q

-Cartier divisor on

X

, then for the sheaf

L = O_{X} (D)

there is a formula of the form

\begin{matrix} (4.1.1) & χ (L) = χ (O_{X}) + \frac{1}{12} D \cdot (D - K_{X}) \cdot (2 D - K_{X}) + \frac{1}{12} D \cdot c_{2} + \sum_{P} c_{P} (D) \end{matrix}

where the sum rungs over all the virtual quotient singularities of

X

, i.e., over the actual singularities of

X

that are replaced with their small deformations [66], and

c_{P} (D)

is a local
contribution due to singularity at

P

, depending only on the local analytic type of

D

P

. There is an explicit formula for the computation of

c_{P} (D)

Except for a few hard cases, the classification of divisorial Mori contractions of a divisor to a point has been completed. A typical result here is to show that a contraction is a weighted blowup with some explicit collection of weights:

Theorem 4.2 (Y. Kawamata [34]). Let

f : X \to (Z ∋ o)

be a divisorial Mori contraction that contracts a divisor to a point. Assume that

o \in Z

is a cyclic quotient singularity of type

\frac{1}{r} (a, - a, 1)

. Then

f

is the weighted blowup with weights

(a / r, 1 - a / r, 1 / r)

Theorem 4.3 (M. Kawakita [26]). Let

f : X \to (Z ∋ o)

be a divisorial Mori contraction that contracts a divisor to a smooth point. Then

f

is the weighted blowup with weights

(1, a, b)

, where

\gcd (a, b) = 1

These results are intensively used in the three-dimensional birational geometry, for example, in the proof of birational rigidity of index-1 Fano threefold weighted hypersurfaces

[11]

5. DEL PEZZO FIBRATIONS

Much less is known about the local structure of

Q

-del Pezzo fibrations. As was explained in Section 3, the existence of a Du Val general elephant would be very helpful in the study such contractions. However, in this case Conjecture 3.1 is established only in some special situations.

An important question that can be asked in the Del Pezzo fibration case is the presence of multiple fibers.

Theorem 5.1 ([49]). Let

f : X \to Z

be a

Q

-del Pezzo fibration and let

f^{*} (o) = m_{o} F_{o}

be a special fiber of multiplicity

m_{o}

. Then

m_{o} \leq 6

and all the cases

1 \leq m_{o} \leq 6

occur. Moreover, the possibilities for the local behavior of

F_{o}

near singular points are described.

The main idea of the proof is to apply the orbifold Riemann-Roch formula (4.1.1) to the divisor

F_{o}

and its multiples.

Example. Suppose that the cyclic group

μ_{4}

acts on

P_{x}^{1} \times P_{y}^{1} \times C_{t}

via

(x, y; t) ⟼ (y, - x, \sqrt{- 1} t)

Then the quotient

X = (P^{1} \times P^{1} \times C) / μ_{4} ⟶ Z = C / μ_{4}

is the germ of a

Q

-del Pezzo fibration with central fiber of multiplicity 4.

Another type of

Q

-del Pezzo fibrations which are investigated relatively well are those whose central fiber

F := f^{- 1} (o)

is reduced, normal, and has "good" singularities. Then

X

can be viewed as a one-parameter smoothing of

F

. The total space of this smoothing must be

Q

-Gorenstein and

F

can be viewed as a degeneration of a general fiber (smooth del Pezzo surface) in a

Q

-Gorenstein family. The most studied class of singularities admitting

Q

-Gorenstein smoothings is the class of singularities of type T.

Definition (KollÃ¡r, Shepherd-Barron [40]). A two-dimensional quotient singularity is said to be of type

T

if it admits a smoothing in a one-parameter

Q

-Gorenstein family

X \to B

In this case, by the inversion of adjunction [70], the pair

(X, F)

is plt and the total family

X

is terminal. Conversely, if

X ∋ P

is a

Q

-Gorenstein point and

F

is an effective Cartier divisor at

P

such that the pair

(X, F)

is plt, then

F ∋ P

is a T-singularity and the point

X ∋ P

is terminal. Singularities of type T and their deformations were studied by KollÃ¡r and Shepherd-Barron [40]. In particular, they proved that any T-singularity is either a Du Val point or a cyclic quotient of type

\frac{1}{m} (q_{1}, q_{2})

with

\gcd (m, q_{1}) = \gcd (m, q_{2}) = 1, {(q_{1} + q_{2})}^{2} \equiv 0 mod m

Minimal resolutions of these singularities are also described [40, Â§ 3].

Thus to study

Q

-del Pezzo fibrations whose central fiber has only quotient singularities, one has to consider

Q

-Gorenstein smoothings of del Pezzo surfaces with singularities of type T. The important auxiliary fact here is the unobstructedness of deformations:

Proposition 5.2 (

[13, 41])

. Let

F

be a projective surface with log canonical singularities such that

- K_{F}

is big. Then there are no local-to-global obstructions to deformations of

F

. In particular, if

F

has

T

-singularities, then

F

admits a

Q

-Gorenstein smoothing.

Theorem 5.3 (Hacking-Prokhorov [13]). Let F be a projective surface with quotient singularities such that

- K_{F}

is ample,

ρ (F) = 1

, and

F

admits a

Q

-Gorenstein smoothing. Then

F

belongs to one of the following:

14 infinite sequences of toric surfaces (see below);
partial smoothing of a toric surface as above;
18 sporadic families of surfaces of index $\leq 2$ [2].

Toric surfaces appearing in the above theorem are determined by a Markov-type equation. More precisely, for

K_{F}^{2} \geq 5

these surfaces are weighted projective spaces given by the following table:

$K_{F}^{2}$	$F$	Markov-type equation
9	$P (a^{2}, b^{2}, c^{2})$	$a^{2} + b^{2} + c^{2} = 3 a b c$
8	$P (a^{2}, b^{2}, 2 c^{2})$	$a^{2} + b^{2} + 2 c^{2} = 4 a b c$
6	$P (a^{2}, 2 b^{2}, 3 c^{2})$	$a^{2} + 2 b^{2} + 3 c^{2} = 6 a b c$
5	$P (a^{2}, b^{2}, 5 c^{2})$	$a^{2} + b^{2} + 5 c^{2} = 5 a b c$

and for

K^{2} \leq 4

they are certain abelian quotients of the weighted projective spaces as above. Note, however, that in general we cannot assert that, for central fiber

F

of a

Q

-del Pezzo fibration, the condition

ρ (F) = 1

holds. Some partial results in the case

ρ (F) > 1

where obtained in [60]. In particular, [60] establishes the existence of Du Val general elephant for

Q

-del Pezzo fibrations with "good" fibers:

Theorem 5.4. Let

f : X \to (Z ∋ o)

be a

Q

-del Pezzo fibration over a curve germ

Z ∋ o

. Assume that the fiber

f^{- 1} (o)

is reduced, normal, and has only log terminal singularities. Then the general elephant

D \in | - K_{X} |

is Du Val.

Theorem 5.3 gives a complete answer to the question posed by M. Manetti [41]:

Corollary 5.5 ([13]). Let

X

be a projective surface with quotient singularities which admits a smoothing to

P^{2}

. Then

X

is a

Q

-Gorenstein deformation of a weighted projective plane

P (a^{2}, b^{2}, c^{2})

, where the triple

(a, b, c)

is a solution of the Markov equation

a^{2} + b^{2} + c^{2} = 3 a b c

Results similar to Theorem 5.3 were obtained for

Q

-del Pezzo fibrations whose central fiber is log canonical [62]. However, in this case the classification is not complete.

6. EXTREMAL CURVE GERMS

To study Mori contractions with fibers of dimension

\leq 1

, it is convenient to work with analytic threefolds and to localize to situation near a curve contained in a fiber.

Definition 6.1. Let

(X \supset C)

be the analytic germ of a threefold with terminal singularities along a reduced connected complete curve. Then

(X \supset C)

is called an extremal curve germ if there exists a contraction

f : (X \supset C) ⟶ (Z ∋ o)

such that

C = f^{- 1} (o)_{red}

and

- K_{X}

f

-ample. The curve

C

is called the central fiber of the germ and

Z ∋ o

is called the target variety or the base of

(X \supset C)

. An extremal curve germ is said to be irreducible if such is its central fiber.

In the definition above, we do not assume that

X

Q

-factorial or

ρ (X / Z) = 1

. This is because

Q

-factoriality typically is not a local condition in the analytic category (see

[32, $ 1])

. There are three types of extremal curve germs:

flipping if $f$ is birational and does not contract divisors;
divisorial if the exceptional locus is two-dimensional;
$Q$ -conic bundle germ if the target variety $Z$ is a surface.

If a divisorial curve germ is irreducible, then the exceptional locus of the corresponding contraction is a

Q

-Cartier divisor and the target variety

Z

has terminal singularities [51, Â§3]. In general, this is not true. It may happen that the exceptional locus is a union of a divisor and some curves.

As an example, we consider the case where

X

has singularities of indices 1 and 2 .

Theorem 6.2 ([47]). Let

(X \supset C)

be a

Q

-conic bundle germ over a smooth base. Assume that

X

is not Gorenstein and

2 K_{X}

is Cartier. Then

X

can be embedded to

P (1, 1, 1, 2) \times C^{2}

and given there by two quadratic equations. In particular, the point

P \in X

of index 2 is unique, the curve

C

has at most 4 components, all of them pass through

P

Theorem 6.3 ([38]). Let

(X \supset C)

be a flipping extremal curve germ and let

be the corresponding flip. Assume that

2 K_{X}

is Cartier. Then

(Z ∋ o)

is the quotient of the isolated hypersurface singularity

{x_{1} x_{3} + x_{2} ϕ (x_{2}^{2}, x_{4}) = 0} ∋ 0

by the

μ_{2}

-action given by the weights

(1, 1, 0, 0)

. The contraction

f

(resp.

f^{+}

) is the quotient of the blowup of the plane

{x_{2} = x_{3} = 0}

(resp.

{x_{1} = x_{2} = 0}

) by

μ_{2}

. In particular,

X

contains a unique point of index 2 and the central fiber

C

is irreducible. The variety

X^{+}

is Gorenstein.

A similar description is known for divisorial extremal curve germs of index 2

[38, 84]

First properties. Let

(X \supset C)

be an extremal curve germ and let

f : (X \supset C) \to (Z ∋ o)

be the corresponding contraction. For any connected subcurve

C^{'} \subset C

, the germ

(X \supset C^{'})

is again an extremal curve germ. If, moreover,

C^{'} ⫋ C

, then

(X \supset C^{'})

is birational. By the Kawamata-Viehweg vanishing theorem,

\begin{matrix} (6.3.1) & R^{1} f_{*} O_{X} = 0 \end{matrix}

(see, e.g., [35]). As a consequence, one has

p_{a} (C^{'}) \leq 0

for any subcurve

C^{'} \subset C

. In particular,

C = ⋃ C_{i}

is a "tree" of smooth rational curves. Furthermore,

\begin{matrix} (6.3.2) & Pic (X) ≃ H^{2} (X, Z) ≃ Z^{\oplus n} \end{matrix}

where

n

is the number of irreducible components of

C

. For more delicate properties of extremal curve germs, one needs to know the cohomology of the dualizing sheaf, see [44,47]:

\begin{matrix} (6.3.3) & R^{1} f_{*} ω_{X} = {\begin{cases} 0, & if f is birational, \\ ω_{Z}, & if f is Q -conic bundle and Z is smooth. \end{cases} \end{matrix}

Definition. An irreducible extremal curve germ

(X \supset C)

is (locally) imprimitive at a point

P

if the inverse image of

C

under the index-one cover

(X^{♯} ∋ P^{♯}) \to (X ∋ P)

splits.

Theorem 6.4 ([44,47]). Let

(X \supset C)

be an extremal curve germ and let

C_{1}, \dots, C_{n}

be irreducible components of

C

Each $C_{i}$ contains at most 3 singular points of $X$ .
Each $C_{i}$ contains at most 2 non-Gorenstein points of $X$ and at most 1 point which is imprimitive for $(X \supset C_{i})$ .
If $X$ is Gorenstein at the intersection point $P = C_{i} \cap C_{j}, C_{i} \neq C_{j}$ , then $X$ is smooth outside $P$ and $(X \supset C)$ is a $Q$ -conic bundle germ over a smooth base.

To prove the first assertion, one needs to analyze the conormal sheaf

I_{C} / I_{C}^{2}

and use the vanishing

H^{1} (O_{X} / J) = 0

for any

J \subset O_{X}

with

Supp (O_{X} / J) = C

(see [44,55]). For the second assertion, one can use topological arguments based on (1.2.1) (see [55]). For the last assertion, we refer to [44, 1.15], [37, 4.2], and [55, 4.7.6]

The techniques applied in the proof of the above proposition allow obtaining much stronger results. In particular, they alow classifying all the possibilities for the local behavior of an irreducible germ

(X \supset C)

near a singular point

P

[44]. Thus, according to [44] and [47], the triple

(X \supset C ∋ P

) belongs to one of the following types:

(IA), (IC), (IIA), (IIB), (IA^{\lor}), ({I I}^{\lor}), ({I D}^{\lor}), ({I E}^{\lor}), (I I I)

Here the symbol

^{\lor}

means that

(X \supset C ∋ P)

is locally imprimitive, the symbol II means that (

X ∋ P

) is a terminal point of exceptional type cAx/4 (see Proposition 3.2), and III means that

(X ∋ P

) is an (isolated) cDV-point.

For example, a triple (

X \supset C ∋ P

) is of type (IC) if there are analytic isomorphisms

(X ∋ P) ≃ C_{y_{1}, y_{2}, y_{4}}^{3} / μ_{m} (2, m - 2, 1), C ≃ {y_{1}^{m - 2} - y_{2}^{2} = y_{4} = 0} / μ_{m}

where

m

is odd and

m \geq 5

. For definitions other types, we refer to [44] and [47].

6.1. Construction of germs by deformations

Let

(X \supset C)

be an extremal curve germ and let

f : X \to (Z ∋ o)

be the corresponding contraction. Denote by

| O_{Z} |

the infinite-dimensional linear system of hyperplane sections passing through

o

and let

| O_{X} | := f^{*} | O_{Z} |

. The general hyperplane section of

(X \supset C

) is the general member

H \in | O_{X} |

. The divisor

H

contains much more information on the total space than a general elephant

D \in | - K_{X} |

. However, the singularities of

H

typically are more complicated, in particular,

H

can be nonnormal.

The variety

X

(resp.

Z

) can be viewed as the total space of a one-parameter deformation of

H

(resp.

H_{Z} := f (H)

). We are going to reverse this consideration.

Construction (see [38, Â§ 11], [44, Â§ 1B]). Suppose we are given a normal surface germ

(H \supset C

) along a proper curve

C

and a contraction

f_{H} : H \to H_{Z}

such that

C

is a fiber and

- K_{H}

f_{H}

-ample. Let

P_{1}, \dots, P_{r} \in H

be all the singular points. Assume also that near each

P_{i}

there exists a small one-parameter deformation

S_{i}

of a neighborhood

H_{i}

P_{i}

H

such that the total space

S_{i}

has a terminal singularity at

P_{i}

. The obstruction to globalize deformations

Ψ : Def (H) ⟶ \prod_{P_{i} \in Sing (H)} Def (H, P_{i})

lies in

R^{2} f_{*} T_{H}

, where

T_{H} = H

(Ω_{H}, O_{H})

is the tangent sheaf of

H

. Since

R^{2} f_{*} T_{H} = 0

due to dimension reasons, the morphism

Ψ

is smooth, and so there exists a global oneparameter deformation

S

H

inducing a local deformation of

S_{i}

near

P_{i}

Then we have a threefold

X := S \supset C

with

H \in | O_{X} |

such that locally near

P_{i}

it has the desired structure and one can extend

f_{H}

to a contraction

f : X \to Z

which is birational (resp. a

Q

-conic bundle) if

H_{Z}

is a surface (resp. a curve).

Example. Consider a rational curve fibration

f_{\tilde{H}} : \tilde{H} \to H_{Z}

over a smooth curve germ

H_{Z} ∋ o

, where

\tilde{H}

is a smooth surface such that the fiber over

o

has the following weighted dual graph:

Contracting the curves corresponding to the white vertices

◻

and

\circ

, we obtain a singular surface

H

and a

K_{H}

-negative contraction

f_{H} : H \to H_{Z}

whose fiber over

o

is a curve

C \subset H

having three irreducible components that correspond to the black vertices

∙

. The singular locus of

H

consists of a Du Val point

P_{0} \in H

of type

A_{1}

and a log canonical singularity

P \in H

whose dual graph is formed by the white circle vertices

\circ

. Both

P_{0}

and

P

have 1-parameter

Q

-Gorenstein smoothings [38, computation 6.7.1]. Applying the above construction to

H \supset C

, we obtain an example of a

Q

-conic bundle contraction

f : (X \supset C) \to (Z ∋ o)

with a unique non-Gorenstein point which is of type cD/3. If we remove the (-2)-curve corresponding to

◻

on the left-hand side of the graph, we get a birational contraction of surfaces

f_{H}^{'} : H^{'} \to H_{Z}^{'}

. Applying the same construction to

H^{'} \supset C

, we obtain an example of a divisorial contraction. Similarly, removing further one of the

(- 1)

-curves, we get a flip.

7. EXTREMAL CURVE GERMS: GENERAL ELEPHANT

Theorem 7.1 (Mori [44], KollÃ¡r-Mori [38], Mori-Prokhorov [50]). Let

(X \supset C)

be an irreducible extremal curve germ. Then the general member

D \in | - K_{X} |

has only Du Val singularities.

The existence of a Du Val elephant for extremal curve germs with reducible central fiber is not known at the moment. See Theorem 9.2 below for partial results in this direction.

Comment on the proof. Essentially, there are three methods to find a good elephant

D \in | - K_{X} |

. We outline them below.

7.1. Local method

As in Proposition 3.2, near each non-Gorenstein point

P_{i} \in X

take a local general elephant

D_{i} \in | - K_{(X ∋ P_{i})} |

. Since

D_{i}

is general, we have

D_{i} \cap C = {P_{i}}

. Then we can regard

D := \sum D_{i}

as a Weil divisor on

X

. By the construction,

K_{X} + D

is a Cartier divisor near each

P_{i}

, hence it is Cartier everywhere. In some cases it is possible to compute the intersection numbers

D_{i} \cdot C

and show that

D \cdot C < 1

. Then we may assume that

K_{X} + D \sim 0

by (6.3.2) and so

D

is a Du Val anticanonical divisor. For example, this method works for extremal curve germs described in Theorems 6.2 and 6.3, as well as in Example 7.3 below.

7.2. Extension from $S \in | - 2 K_{X} |$

In some cases, the above approach does not work, but it allows showing the existence of a klt 2-complement

S \in | - 2 K_{X} |

such that

\dim (D \cap C) = 0

. Then one can try to extend a good element from the surface

S

. The crucial fact here is that the natural map

τ : H^{0} (X, O_{X} (- K_{X})) ⟶ H^{0} (S, O_{S} (- K_{X})) = ω_{S}

is surjective if

(X, C)

is birational and surjective modulo

Ω_{S}^{2}

(X, C)

is a

Q

-conic bundle. This immediately follows from (6.3.3). Details can be found in [38, Â§ 2] and [50].

7.3. Global method

Finally, in the most complicated cases, none of the above methods work. Then one needs more subtle techniques which require detailed analysis of singularities and infinitesimal structure of

X

along

C

[44, Â§Â§ 8-9]. Then, roughly speaking, the good section

D \in | - K_{X} |

is recovered as the formal Weil divisor

lim C_{n}

of the completion

X^{\land}

X

along

C

, where

C_{n}

are subschemes with support

C

constructed by using certain inductive procedure

[44, § 9]

As a consequence of Theorem 7.1, in the

Q

-conic bundle case, one obtains the following fact which proves Iskovskikh's conjecture [24].

Corollary 7.2. Let

(X \supset C)

be a

Q

-conic bundle germ over

(Z ∋ o)

, where

C

can be reducible. Then

(Z ∋ o)

is a Du Val singularity of type

A_{n}

(or smooth).

This result is very useful for applications to the rationality problem of three-dimensional varieties having conic bundle structure

[24, 61]

and some problems of biregular geometry

[58, 59]

It turns out that the structure of

Q

-conic bundle germs over a singular base (

Z ∋ o

) is much simpler and shorter than others. In fact, these germs can be exhibited as certain quotients of

Q

-conic bundles of index

\leq 2

(see Theorem 6.2). A complete classification of such germs was obtained in

[47, 48]

. Here is a typical example.

Example 7.3. Let the group

μ_{n}

act on

C_{u, v}^{2}

and

P_{x, y}^{1} \times C_{u, v}^{2}

via

(x : y; u, v) ⟼ (x : ζ^{a} y; ζ u, ζ^{- 1} v)

where

ζ = ζ_{n} = \exp (2 π i / n)

and

\gcd (n, a) = 1

. Then the projection

f : X = (P^{1} \times C^{2}) / μ_{n} ⟶ Z = C^{2} / μ_{n}

is a

Q

-conic bundle. The variety

X

has exactly two singular points which are terminal cyclic quotients of type

\frac{1}{n} (1, - 1, \pm a)

. The surface

Z

has at 0 a

D u

Val point of type

A_{n - 1}

McKernan proposed a natural higher-dimensional analogue of Corollary 7.2:

Conjecture 7.4. Let

f : X \to Z

be a

K

-negative contraction such that

ρ (X / Z) = 1

and

X

ε

-lc, that is, all the coefficients in (1.1.1) satisfy

a_{i} \geq - 1 + ε

. Then

Z

δ

-lc, where

δ

depends on

ε

and the dimension.

A deeper version of this conjecture which generalizes Theorem 5.1 and uses the notion was proposed by Shokurov. He also suggested that the optimal value of

δ

, in the
case where singularities of

X

are canonical and

f

has one-dimensional fibers, equals

1 / 2

. Recently, this was proved by J. Han, C. Jiang, and Y. Luo [17].

Once we have a Du Val general elephants, all extremal curve germs can be divided into two large classes which will be discussed separately in the next two sections.

Definition 7.5. Let

(X \supset C)

be an extremal curve germ and let

f : X \to (Z ∋ o)

be the corresponding contraction. Assume that the general member

D \in | - K_{X} |

is Du Val. Consider the Stein factorization:

f_{D} : D ⟶ D^{'} ⟶ f (D) (put D^{'} = f (D) if f is birational).

Then the germ

(X \supset C)

is said to be semistable if

D^{'}

has only (Du Val) singularities of type

A_{n}

. Otherwise,

(X \supset C)

is called exceptional.

8. SEMISTABLE GERMS

Let

(X \supset C)

be an irreducible extremal curve germ. By Theorem 7.1, the general member

D \in | - K_{X} |

is Du Val. In this section we assume that

(X \supset C)

is semistable. Excluding simple cases, we assume also that

X

is not Gorenstein [12] and

(X \supset C)

is not a

Q

-conic bundle germ over a singular base

[47, 48]

. According to Theorem 6.4, the threefold

X

has at most two non-Gorenstein points. Thus the following case division is natural:

Case (k1A): the set of non-Gorenstein points consists of a single point

P

;

Case (k2A): the set of non-Gorenstein points consists of exactly two points

P_{1}, P_{2}

Proposition 8.1. In the above hypothesis, for the general member

H \in | O_{X} |

, the pair

(X, H + D)

is lc. If, moreover,

D \supset C

, then

H

is normal and has only cyclic quotient singularities. In this case the singularities of

H

are of type

T

The proof uses the inversion of adjunction [70] to extend a general hyperplane section from

D

X

(see [51, PROPOSITION 2.6]).

For an extremal curve germ of type (k2A), any member

D \in | - K_{X} |

contains

C

[38]. Hence the general hyperplane section

H \in | O_{X} |

has only T-singularities and

X

can be restored as a one-parameter deformation space of

H

. In this case

X

has no singularities other than

P_{1}, P_{2}

. Moreover,

(X \supset C)

cannot be a

Q

-conic bundle germ [47,50]. The birational germs of type (k2A) were completely described by Mori [46]. He gave an explicit algorithm for computing divisorial contractions and flips in this case.

The structure of extremal curve germs of type (k1A) is more complicated. They were studied in [51]. In particular, the general hyperplane section

H \in | O_{X} |

was computed. However, [51] does not provide a good description of the infinitesimal structure of

X

along

C

or an algorithm similar to [46]. This was done only in a special situation in [14]. Note that in the case

(k 1 A)

a general member

H \in | O_{X} |

can be nonnormal.

Examples. Similar to the example in Section 6.1, consider a surface germ

H \supset C ≃ P^{1}

whose dual graph has the following graph of the minimal resolution:

- 7

- 2

- 2

- 2

where

∙

is a ( -1 -curve. The chain formed by white circle vertices o corresponds to a Tsingularity of type

\frac{1}{25} (1, 4)

. The whole configuration can be contracted to a cyclic quotient singularity

H_{Z} ∋ o

of type

\frac{1}{21} (1, 16)

. Since this is not a T-singularity, the induced threefold contraction must be flipping.

9. EXCEPTIONAL CURVE GERMS

In this section we assume that

(X \supset C)

is an exceptional irreducible extremal curve germ. As in the previous section we also assume that

X

is not Gorenstein and

(X \supset C)

is not a

Q

-conic bundle germ over a singular base. According to the classification

[38, 44, 50]

, the germ

(X \supset C)

belongs to one of following types:

$X$ has a unique non-Gorenstein point $P$ which is of type cD/2, cAx/2, cE/2, or $c D / 3$ and $(X \supset C)$ is of type (IA) at $P$ ;
$X$ has a unique non-Gorenstein point $P$ which is of exceptional type $c A x / 4$ and $(X \supset C)$ is of type (IIA), (II $^{\lor})$ , or (IIB) at $P$ ;
$X$ has a unique singular point $P$ which is a cyclic quotient singularity of index $m \geq 5$ (odd) and $(X \supset C)$ is of type (IC) at $P$ ;
$X$ has two singular points of indices $m \geq 3$ (odd) and 2, then $(X \supset C)$ is said to be of type (kAD);
$X$ has three singular points of indices $m \geq 3$ (odd), 2 and 1 , then $(X \supset C)$ is said to be of type $(k 3 A)$ .

In each case the general elephant is completely described in terms of its minimal resolution:

Theorem 9.1 ([38,50]). In the above hypothesis assume that the general element

D \in | - K_{X} |

contains

C

. Then the dual graph of

(D \supset C)

is one of the following, where white vertices

\circ

denote (-2)-curves on the minimal resolution of

D

and the black vertex

∙

corresponds to the proper transform of

C

(IC)

(k A D)

(k3A)

0 - \dots - 0 - \begin{matrix} 0 \\ 1 \\ 0 \end{matrix}]

Exceptional irreducible extremal curve germs are well studied (see

[38, 55]

, and references therein). For flipping ones, the general hyperplane section

H \in | O_{X} |

is normal and has only rational singularities. It is computed in [38] and the flip is reconstructed as a oneparameter deformation space of

H

. For divisorial and

Q

-conic bundle germs, the situation is more complicated. Then the general hyperplane section

H

can be nonnormal (see, e.g., [54]). Nevertheless, in almost all cases, except for types (kAD) and (k3A), there is a description of

H \in | O_{X} |

and infinitesimal structure of these germs. For convenience of reference, in the table below we collect the known information on the exceptional irreducible extremal curve germs.

Type	$(X, C)$	References
index 2 germs	divisorial, $Q$ -conic bundle	$[38, § 4], [47, § 12], [51, § 7]$
cD/3	flip, divisorial	$[38, § 6], [51, § 4]$
(IC)	flip, $Q$ -conic bundle (only for $m = 5)$	$[38, 88], [52]$
(IIA)	flip, divisorial, $Q$ -conic bundle	$[38, § 7], [53, 54]$
(IIB)	divisorial, $Q$ -conic bundle	$[52]$
(II $^{\lor})$	divisorial, $Q$ -conic bundle	$[38, 4.11 .2], [47]$
(kAD)	flip, divisorial, $Q$ -conic bundle	$[38, § 9], [45, 47, 50]$
(k3A)	divisorial, $Q$ -conic bundle	$[38, § 5], [47, 50]$

Detailed analysis of the local structure of exceptional extremal curve germs allows extending the result of Theorem 7.1 to the case of reducible central fiber containing an exceptional component:

Theorem 9.2 (Mori-Prokhorov [56]). Let

(X \supset C)

be an extremal curve germ such that

C

is reducible and satisfies the following condition:

^{(*)}

each component

C_{i} \subset C

contains at most one point of index

> 2

Then the general member

D \in | - K_{X} |

has only Du Val singularities. Moreover, for each irreducible component

C_{i} \subset C

with two non-Gorenstein points or of types (IC) or (IIB), the dual graph of

(D, C_{i})

has the same form as the irreducible extremal curve germ

(X \supset C_{i})

The proof uses the extension techniques of sections of

| - K_{X} |

from a good member

S \in | - 2 K_{X} |

(see Section 7.2).

10. $Q$ -FANO THREEFOLDS

In arbitrary dimension,

Q

-Fano threefolds are bounded, i.e., they are contained in fibers of a morphism of schemes of finite type. This is a consequence of the much more general fact [4]. In dimension 3, there are effective results based on the orbifold RiemannRoch formula (4.1.1) and Bogomolov-Miyaoka inequality applied to the restriction of the
reflexive sheaf

{(Ω_{X}^{1})}^{\lor \lor}

to a sufficiently general hyperplane section [33]. In particular, combining (4.1.1) with Serre duality, we obtain

χ (O_{X}) = \frac{1}{24} (- K_{X} \cdot c_{2} (X) + \sum_{P} (m_{P} - \frac{1}{m_{p}}))

where

m_{P}

is the index of a virtual quotient singularity of

X

[66]. Since

X

Q

-Fano, by Kawamata-Viehweg vanishing theorem [35], one has

χ (O_{X}) = 1

. Arguments based on Bogomolov-Miyaoka inequality show that

- K_{X} \cdot c_{2} (X)

is positive (see [33]). This gives an effective bound of the indices of singularities of

X

. Similarly, one can get an upper bound of the anticanonical degree

- K_{X}^{3}

. Moreover, analyzing the methods of [33], it is possible to enumerate Hilbert series of all

Q

-Fano threefolds. This information is collected in [6] in a form of a huge computer database of possible "candidates." It was extensively explored by many authors, basically to obtain lists of examples representing

Q

-Fano threefolds as subvarieties of small codimension in a weighted projective space (see, e.g.,

[7, 21]

and references therein).

Examples. - There are 130 (resp. 125) families of

Q

-Fano threefolds that are representable as hypersurfaces (resp. codimension 2 complete intersections) in weighted projective spaces

[6, 21]

Toric $Q$ -Fano threefolds are exactly weighted projective spaces $P (3, 4, 5, 7)$ , $P (2, 3, 5, 7), P (1, 3, 4, 5), P (1, 2, 3, 5), P (1, 1, 2, 3), P (1, 1, 1, 2), P^{3} = P (1, 1$ , $1, 1)$ , and the quotient of $P^{3}$ by $μ_{5}$ that acts diagonally with weights $(1, 2, 3, 4) [6]$ .

Although the classification is very far from completion, there are several systematic results. For example, the optimal upper bound of the degree

- K_{X}^{3}

Q

-Fano threefolds was obtained in [58]. If

X

is singular, it is equal to 125/2 and achieved for the weighted projective space

P (1, 1, 1, 2)

. The lower bound of the degree equals

1 / 330

[8] and is achieved for a hypersurface of degree 66 in

P (1, 5, 6, 22, 33)

. It is known that, under certain conditions, General Elephant Conjecture 3.1 holds for

Q

-Fano threefolds modulo deformations [67].

ACKNOWLEDGMENTS

The author would like to thank Professors Shigefumi Mori and Vyacheslav Shokurov for helpful comments on the original version of this paper.

FUNDING

This work was performed at the Steklov International Mathematical Center and supported by the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-15-2019-1614).

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YURI PROKHOROV

Steklov Mathematical Institute, 8 Gubkina street, Moscow 119991, Russia, prokhoro @mi-ras.ru

SOME ASPECTS OF RATIONAL POINTS AND RATIONAL CURVES

OLIVIER WITTENBERG

ABSTRACT

Various methods have been used to construct rational points and rational curves on rationally connected algebraic varieties. We survey recent advances in two of them, the descent and the fibration method, in a number-theoretical context (rational points over number fields) and in an algebro-geometric one (rational curves on real varieties), and discuss the question of rational points over function fields of

p

-adic curves.

MATHEMATICS SUBJECT CLASSIFICATION 2020

Primary 11G35; Secondary 14G05, 14C25, 14P99

KEYWORDS

Rational points, rational curves, inverse Galois problem, tight approximation,

p

-adic function fields

1. INTRODUCTION

Let

X

be an algebraic variety over a field

k

and

X (k)

the set of its rational points.

The search for explicit descriptions of the set

X (k)

when

k

is a number field is one of the oldest themes of number theory. A modern point of view on this problem consists in embedding

X (k)

diagonally into the topological space

X (A_{k})

of adelic points of

X

and attempting to identify its topological closure. By general principles that were formulated by Lang after the works of Mordell, Weil, and Siegel, the answer is expected to depend in a crucial manner on the geometry of

X

. For instance, assuming that

X

is smooth and projective and that an embedding

k ↪ C

is given, the set

X (k)

is conjectured to be finite if the complex variety

X_{C}

is hyperbolic (see [71]). One may then seek to count, list, or bound its elements. At the other end of the spectrum, if

X_{C}

is a rationally connected smooth projective variety in the sense of Campana [12] and KollÃ¡r-Miyaoka-Mori [69], then one expects that the set

X (k)

is Zariski dense in

X

whenever it is nonempty; more precisely, by a conjecture of Colliot-ThÃ©lÃ¨ne, the closure of

X (k)

X (A_{k})

should coincide in this case with the BrauerManin set

X {(A_{k})}^{Br (X)}

defined by Manin [76]. This far-reaching conjecture encompasses in particular the inverse Galois problem, and its refinement the Grunwald problem (see

[28, 33

36]

[95, $ 3.5]

Criteria for the existence of rational points on

X

are also of relevance outside of number theory, when

k

is no longer assumed to be a number field. For instance, the GraberHarris-Starr theorem [34], a central result in the theory of rational curves on complex algebraic varieties, is equivalent to the statement that

X (k) \neq \emptyset

k

is the function field of a complex curve and

X

is a rationally connected variety. (We say that

X

is rationally connected to mean that for any algebraically closed field extension

K

k

, the variety

X_{K}

over

K

is rationally connected in the sense of

[12, 69]

.) As another example, if

X

is a real algebraic variety with no real point and

k

denotes the function field of the real conic given by

x^{2} + y^{2} = - 1

, the existence of a geometrically rational curve on

X

-a property conjectured by KollÃ¡r to hold whenever

X

is a positive-dimensional rationally connected variety-is equivalent to the statement that

X (k) \neq \emptyset

The results we discuss in this expository article concern the existence of rational points in two very distinct contexts, leading to the following two concrete theorems, obtained in collaboration with Yonatan Harpaz and with Olivier Benoist, respectively. As we shall see, their proofs roughly follow, perhaps somewhat surprisingly, a common general strategy.

Theorem A (see [50]). Let

G

be a finite nilpotent group. Let

k

be a number field.

(1) There exist Galois extensions

K / k

with Galois group

G

(2) If

v_{1}, \dots, v_{n}

are pairwise distinct places of

k

none of which is a finite place dividing the order of

G

, and

w_{1}, \dots, w_{n}

are places of

K

above

v_{1}, \dots, v_{n}

, then in (1), one can require that the extensions

K_{w_{i}} / k_{v_{i}}

be isomorphic to any prescribed collection of Galois extensions of

k_{v_{1}}, \dots, k_{v_{n}}

whose Galois groups are subgroups of

G

Theorem B (see [6]). Let

X

be a smooth, proper variety over

R

. Let

ε : S^{1} \to X (R)

be a continuous map. Assume that

X

is birationally equivalent to a homogeneous space of a linear algebraic group over

R

. Then there exist morphisms of algebraic varieties

P_{R}^{1} \to X

that induce maps

P^{1} (R) = S^{1} \to X (R)

arbitrarily close to

ε

in the compact-open topology.

Theorem A (1) was first proved by Shafarevich in his seminal work on the inverse Galois problem for solvable groups (see [82, CHAPTER IX, Â§ 6]; it should be noted that nilpotent groups form the most difficult case in his proof); the proof given in [50] is independent from his and has a geometric flavour. Theorem A (2), on the other hand, was new in [50] and was not accessible with Shafarevich's methods.

As far as we know, Theorem B might hold under the sole assumption that

X

is rationally connected. This is a question we raise in [6]. Theorem B provides the first examples of a positive answer to it for varieties that are not

R

-rational (indeed, not even

C

-rational). For

R

-rational varieties, the conclusion of Theorem B was previously shown, by Bochnak and Kucharz [8], to follow from the Stone-Weierstrass theorem.

The first step in the proofs of Theorems A and B consists in strengthening and reformulating the desired conclusion in terms of the existence of suitable rational points on suitable varieties over suitable fields. In the case of Theorem A, the varieties in question are homogeneous spaces of

{S L}_{n}

over number fields; for the proof, though not for the statement, it is crucial to not restrict to homogeneous spaces that have rational points (i.e., to homogeneous spaces of the form

{S L}_{n} / G)

. In the case of Theorem B, the varieties in question are homogeneous spaces of linear algebraic groups, over the rational function field

R (t)

; for the proof, though not for the statement, it is crucial to not restrict to homogeneous spaces or algebraic groups that are defined over

R

. In the remainder of the proofs of Theorems A and

B

, one establishes the validity of these strengthened formulations by combining geometric dÃ©vissages of the underlying algebraic varieties with two general tools: the descent method and the fibration method. The fibration method, whose first instance can be found in the work of Hasse on the local-global principle for quadratic forms, consists in reducing the desired property for a variety

V

endowed with a morphism

p : V \to B

with geometrically irreducible generic fibre to the same property for

B

and for a collection of smooth fibres of

p

. The descent method, which goes back to Fermat, attempts to reduce the desired property for a variety

V

endowed with a torsor

p : W \to V

under a (possibly disconnected) linear algebraic group over

k

to the same property for

W

and for all of its twists. It was developed in the context of elliptic curves, for torsors under finite abelian groups, by Mordell, Cassels, and Tate, and the setup was later extended to torsors under positive-dimensional linear algebraic groups by Colliot-ThÃ©lÃ¨ne and Sansuc, Skorobogatov, Harari.

We take Theorems A and B as excuses leading us to the general study of rational points on rationally connected varieties defined over number fields or over function fields of real curves. We discuss recent advances in the fibration and descent methods in these two contexts in Sections 2 and 3, stating along the way the main open questions that surround Theorems A and B and their proofs. We then turn, in Section 4, to function fields of

p

-adic curves, and speculate about the existence of a

p

-adic analogue of the "tight approximation"
property discussed in Section 3 that would enable one to exploit fibration and descent methods in the study of rational curves over

p

-adic fields and more generally of rational points over function fields of

p

-adic curves.

2. SOLVABLE GROUPS AND THE GRUNWALD PROBLEM IN INVERSE GALOIS THEORY

2.1. Homogeneous spaces

It is the following general theorem about the arithmetic of homogeneous spaces of linear algebraic groups that underlies Theorem A.

Theorem 2.1. Let

V

be a homogeneous space of a connected linear algebraic group

L

over a number field

k

. Let

X

be a smooth compactification of

V

. Let

\bar{v} \in V (\bar{k})

. Assume that the group of connected components

G

of the stabiliser of

\bar{v}

is supersolvable, in the sense that it possesses a normal series

1 = G_{0} ◃ \dots ◃ G_{m} = G

such that the quotients

G_{i + 1} / G_{i}

are cyclic while the subgroups

G_{i}

are normal in

G

and are stable under the natural outer action of

Gal (\bar{k} / k)

G

. Then the subset

X (k)

is dense in

X {(A_{k})}^{Br (X)}

Here and elsewhere, by "compactification of

V

," we mean a proper variety over

k

that contains

V

as a dense open subset; we do not require that the algebraic group

L

act on the compactification. Examples of supersolvable groups with respect to the trivial outer action of

Gal (\bar{k} / k)

include finite nilpotent groups and dihedral groups. With a nontrivial outer action of

Gal (\bar{k} / k)

, however, even abelian groups need not be supersolvable. Previous work of Borovoi [10] nevertheless establishes the conclusion of Theorem 2.1 in many cases where the stabiliser of

\bar{v}

is abelian but not necessarily supersolvable.

Theorem 2.1 can be found in [50, THÃ‰ORÄ–ME B] in the particular case where

L

is semi-simple simply connected and the stabiliser of

\bar{v}

is finite, and in [51, coroLLARY 4.5] in general. To deduce Theorem A from it, embed

G

into

{S L}_{n} (k)

for some

n

, take

L = {S L}_{n}

and

V = {S L}_{n} / G

and let

H

denote the set of points of

V

above which the fibre of the Ã©tale cover

π : L \to V

is irreducible. The function field of the fibre of

π

above any rational point contained in

H

is a Galois extension of

k

with Galois group

G

. On the other hand, by a theorem of Ekedahl [33], the density of

X (k)

X {(A_{k})}^{Br (X)}

implies that of

X (k) \cap H

X {(A_{k})}^{Br (X)}

. Thus, Theorem 2.1 ensures the existence of Galois extensions

K / k

with Galois group

G

having a local behaviour prescribed by any element of the Brauer-Manin set

X {(A_{k})}^{Br (X)}

; that is, one may freely prescribe the completions of

K

at any finite set of places of

k

, as long as these prescriptions satisfy a certain global reciprocity condition determined by

Br (X)

. By a theorem of Lucchini Arteche [75, 86 6, this reciprocity condition imposes, in fact, no restriction at the places indicated in Theorem A (2).

2.2. Geometry

In the special case where

L = {S L}_{n}

and the stabiliser of

\bar{v}

is a finite group

G

, the geometry behind the proof of Theorem 2.1 can be summarised with the following assertion:
there exist an algebraic torus

T

over

k

and a torsor

\bar{Y} \to X_{\bar{k}}

under

T_{\bar{k}}

whose isomorphism class is invariant under

Gal (\bar{k} / k)

, such that for any torsor

Y \to X

under

T

whose base change to

X_{\bar{k}}

is isomorphic to

\bar{Y}

, there exist a dense open subset

W \subseteq Y

and a smooth morphism

p : W \to Q

to a quasitrivial torus

Q

(i.e., a torus of the form

R_{E / k} G_{m}

for a nonzero Ã©tale

k

-algebra

E

) whose fibres are homogeneous spaces of

{S L}_{n}

with geometric stabiliser isomorphic to

G_{m - 1}

. In addition, the morphism

p

admits a rational section over

\bar{k}

This geometry is the key to a proof of Theorem 2.1 by an induction on

m

, at each step of which one applies the descent method and the fibration method, in the form of Theorems 2.2 and 2.3 below. It should be noted that even if

G

is embedded into

{S L}_{n} (k)

and

V = {S L}_{n} / G

, the homogeneous spaces of

{S L}_{n}

that arise as fibres of

p

need not possess rational points. Thus, for the induction to be possible, one cannot restrict the statement of Theorem 2.1 to homogeneous spaces of the form

{S L}_{n} / G

, even though only homogeneous spaces of this form are relevant for Theorem A.

2.3. Descent

The following theorem, which was established in [50] and can also be deduced from [13], is the definitive statement of descent theory in the case of smooth and proper rationally connected varieties over number fields. For geometrically rational

X

, this theorem is due to Colliot-ThÃ©lÃ¨ne and Sansuc [25]. The homogeneous spaces of Theorem 2.1 are not geometrically rational in general (Saltman, Bogomolov; see [26]).

Theorem 2.2. Let

X

be a smooth and proper rationally connected variety over a number field

k

. Let

T

be a torus over

k

and

\bar{Y} \to X_{\bar{k}}

a torsor under

T_{\bar{k}}

whose isomorphism class is invariant under

Gal (\bar{k} / k)

. Then

X {(A_{k})}^{Br (X)} = ⋃_{f : Y \to X} f^{'} (Y^{'} {(A_{k})}^{Br (Y^{'})})

where the union ranges over the torsors

f : Y \to X

under

T

whose base change to

X_{\bar{k}}

is isomorphic to

\bar{Y}

, and

Y^{'}

denotes a smooth compactification of

Y

such that

f

extends to a morphism

f^{'} : Y^{'} \to X

. In particular, if

Y^{'} (k)

is dense in

Y^{'} {(A_{k})}^{Br (Y^{'})}

for every such

f

, then

X (k)

is dense in

X {(A_{k})}^{Br (X)}

(To bridge the gap between Theorem 2.2 and [50, THÃ‰ORÃˆME 2.1], one needs to know that

X {(A_{k})}^{Br (X)} \neq \emptyset

implies the existence of at least one

f

. This goes back to [25] and follows from [102, THEOREM 3.3.1], [25, PROPOSITION 2.2.5], [103, (3.3)].)

2.4. Fibration

The following fibration theorem suffices for the proof of Theorem 2.1. It results from combining a descent with the work of Harari [35] on the fibration method.

Theorem 2.3. Let

p : Z \to B

be a dominant morphism between irreducible, smooth, and proper varieties over a number field

k

, with rationally connected generic fibre. Assume that

(1) there exist dense open subsets

W \subset Z

and

Q \subset B

such that

Q

is a quasitrivial torus over

k

and

p

induces a smooth morphism

W \to Q

with geometrically irreducible fibres;

(2) the morphism

p

admits a rational section over

\bar{k}

;

(3) for all

b \in B (k)

in a dense open subset of

B

, the set

Z_{b} (k)

is dense in

Z_{b} {(A_{k})}^{Br (Z_{b})}

Then

Z (k)

is dense in

Z {(A_{k})}^{Br (Z)}

The assumptions of Theorem 2.3 imply that

B

k

-rational. Under the condition that

B

k

-rational, the first two assumptions of Theorem 2.3 are expected to be superfluous (even under weaker hypotheses on the generic fibre of

p

than rational connectedness, see [48, COROLLARY 9.23 (1)-(2)]), but removing them altogether is a wide-open problem, well connected with analytic number theory (see [48, Â§ 9], [47]). Removing (2) while keeping (1) might be within reach, though:

Question 2.4. In the statement of Theorem 2.3, can one dispense with the assumption that

p

admit a rational section over

\bar{k}

This would allow one to replace "supersolvable" with "solvable" in the statement of Theorem 2.1. Indeed, in Section 2.2, the cyclicity of the quotient

G_{m} / G_{m - 1}

plays a rÃ´le only to ensure the existence of a rational section of

p

over

\bar{k}

(see [50, PROPOSITION 3.3 (II)]).

2.5. An application to Massey products

Theorem 2.1 has concrete applications, over number fields, beyond the inverse Galois problem: for the homogeneous spaces that appear in its statement, it turns the problem of deciding the existence of a rational point into the much more approachable question of deciding the non-vacuity of the Brauer-Manin set. In this way, Theorem 2.1 can be used to confirm, in the case of number fields, the conjecture of MinÃ¡Ä and TÃ¢n on the vanishing of Massey products in Galois cohomology (see [49]). Indeed, this conjecturewhich posits that for any field

k

, any prime number

p

, any integer

m \geq 3

and any classes

a_{1}, \dots, a_{m} \in H^{1} (k, Z / p Z)

, the

m

-fold Massey product of

a_{1}, \dots, a_{m}

vanishes if it is defined (see [78,79])â€”can be reinterpreted, according to PÃ¡l and Schlank [83], in terms of the existence of rational points on appropriate homogeneous spaces of

{S L}_{n}

over

k

(with

n ≫ 0

), and it so happens that the geometric stabilisers of these homogeneous spaces are finite and supersolvable.

3. RATIONAL CURVES ON REAL ALGEBRAIC VARIETIES

3.1. A few questions

Let

X

be a smooth variety over

R

. The interplay between the topology of the

C^{\infty}

manifold

X (R)

and the geometry of the algebraic variety

X

lies at the core of classical
real algebraic geometry. One of the fundamental problems in this area consists in investigating which submanifolds of

X (R)

can be approximated, in the Euclidean topology, by Zariski closed submanifolds. Even for 1-dimensional submanifolds, i.e., disjoint unions of

C^{\infty}

loops, various phenomena-of a topological, Hodge-theoretic, or yet more subtle nature-can obstruct the existence of algebraic approximations (see [4, Â§ 4]). In the case of 1-dimensional submanifolds, however, all known obstructions vanish when

X

is rationally connected. One can thus raise the following questions, in which

H_{1}^{alg} (X (R), Z / 2 Z)

denotes the image of the cycle class map

{C H}_{1} (X) \to H_{1} (X (R), Z / 2 Z)

defined by Borel and Haefliger [9].

Questions 3.1. Let

X

be a smooth, proper, rationally connected variety, over

R

(1) Can all

C^{\infty}

loops in

X (R)

be approximated, in the Euclidean topology, by real loci of algebraic curves? or even by real loci of rational algebraic curves?

(2) Is

H_{1} (X (R), Z / 2 Z) = H_{1}^{alg} (X (R), Z / 2 Z)

? Is

H_{1} (X (R), Z / 2 Z)

generated by classes of rational algebraic curves on

X

The first parts of Questions 3.1 (1) and (2) are in fact equivalent to each other, by the work of Akbulut and King (see [5, Ñ‚HEOREM 6.8]), and were studied in a systematic fashion in [4,5]. The second part of Question 3.1 (1) is, however, as far as we know, genuinely stronger than the second part of Question 3.1 (2). We note that in order to formulate the second part of Question 3.1 (1) precisely, it is better to work with possibly noninjective

C^{\infty}

maps

P^{1} (R) \to X (R)

rather than with submanifolds of

X (R)

. Indeed, there are examples of

R

-rational surfaces

X

and of

C^{\infty}

loops in

X (R)

such that the desired rational algebraic curves necessarily have singular real points (see [68, THEOREM 3]).

A specific motivation for Question 3.1 (2) is its analogy with the following questions in complex geometry raised by Voisin [101] and by KollÃ¡r [67]:

Questions 3.2. Let

X

be a smooth, proper, rationally connected variety, over

C

. Is the group

H_{2} (X (C), Z)

generated by homology classes of algebraic curves? Is it generated by homology classes of rational algebraic curves?

The two parts of Questions 3.2 are in fact equivalent: Tian and Zong [100] have shown that the homology class of any algebraic curve on a rationally connected variety over

C

is a linear combination of homology classes of rational curves. The real analogue of their result remains unknown in general. Its validity is an interesting open problem.

The first parts of Questions 3.1 (2) and of Questions 3.2 are in fact related by more than an analogy: if

X

is a smooth, proper, rationally connected variety over

R

such that

X (R) \neq \emptyset

and such that Questions 3.2 admit a positive answer for

X_{C}

, then the equality

H_{1} (X (R), Z / 2 Z) = H_{1}^{alg} (X (R), Z / 2 Z)

is equivalent to the real integral Hodge conjecture for 1-cycles on

X

, a property formulated and studied in [4,5].

In a different line of investigation around the abundance of rational curves on rationally connected varieties, many authors have considered the problem of finding rational curves through a prescribed set of points, or more generally through a prescribed curvilinear
0-dimensional subscheme, on any smooth, proper, rationally connected variety

X

. Over the complex numbers, such curves exist unconditionally (KollÃ¡r, Miyaoka, Mori, see [63, chAPTER IV.3]). Over the real numbers, such curves exist under the necessary condition that all the prescribed points that are real belong to the same connected component of

X (R)

(KollÃ¡r, see

[64, 66])

. This problem can be generalised to one-parameter families: given a morphism

f : X \to B

with rationally connected generic fibre between smooth and proper varieties, where

B

is a curve, one looks for sections of

f

whose restriction to a given 0 -dimensional subscheme of

B

is prescribed, thus leading to Questions 3.3 below. For simplicity of notation, in the statement of Question 3.3 (2), this 0 -dimensional subscheme of

B

is assumed to be reduced; there is, however, no loss of generality in doing this, since jets of sections can be prescribed at any higher order by replacing

X

with a suitable iterated blow-up (see [52, PROPOSITION 1.4]).

Questions 3.3. Let

B

be a smooth, proper, connected curve over a field

k_{0}

. Let

X

be a smooth, proper variety over

k_{0}

, endowed with a flat morphism

f : X \to B

with rationally connected generic fibre. Let

P \subset B

be a reduced 0-dimensional subscheme. Let

s : P \to X

be a section of

f

over

P

(1) If

k_{0} = C

, can

s

be extended to a section of

f

(2) If

k_{0} = R

and the map

{s |}_{P (R)} : P (R) \to X (R)

can be extended to a

C^{\infty}

section of

{f |}_{X (R)} : X (R) \to B (R)

, can then

s

be extended to a section of

f

Let

X

be the generic fibre of

f

and

k

the function field of

B

. The existence of sections extending any given

s

as above is equivalent to the density of

X (k)

in the topological space

X (A_{k}) = \prod_{b} X (k_{b})

of adelic points of

X

, where the product runs over the closed points

b

B

and

k_{b}

denotes the completion of

k

b

. This is the weak approximation property.

The Graber-Harris-Starr theorem [34] provides a positive answer to Question 3.3 (1) when

P = \emptyset

and it is a conjecture of Hassett and Tschinkel that the answer to this question is in the affirmative in general (see

[17, 52, 53, 99]

for known results). Particular cases of Question 3.3 (2) were first studied by Colliot-ThÃ©lÃ¨ne [14], who conjectured the validity of weak approximation (i.e., a positive answer to Question 3.3 (2) even without assuming that

{s |}_{P (R)}

can be extended to a

C^{\infty}

section of

{f |}_{X (R)})

when

X

is birationally equivalent to a homogeneous space of a connected linear algebraic group over

k

, and proved his conjecture when the geometric stabilisers are trivial. Scheiderer [94] then proved the same conjecture when the geometric stabilisers are connected. Ducros [30,31] stated Question 3.3 (2) in these exact terms, and gave a positive answer when

X

is a conic bundle surface, or more generally when there exists a dominant map

X \to P_{k}^{1}

whose generic fibre is a Severi-Brauer variety.

3.2. Tight approximation

The main insight behind the proof of Theorem B is the observation that formulating a suitable common strengthening of Questions 3.1 and 3.3, through the notion of tight approximation, can render all of these questions fully amenable to both the descent method
and the fibration method. We note that Questions 3.1 and Questions 3.3 are somewhat orthogonal in spirit, insofar as the former consider global constraints on curves lying on

X

, while the latter are aimed at local constraints.

The idea of establishing a descent method (resp. fibration method) for Question 3.3 (2) already appeared in [31] (resp. [84]), though in [31] and [84] the implementations are subject to miscellaneous restrictions. The possibility of a descent method and a fibration method for studying Questions 3.1, however, is new and turns out to require a shift in perspective from single rationally connected varieties to one-parameter families of such.

Let us illustrate how Questions 3.1 need to be strengthened for a fibration argument to go through. We start with a dominant morphism

p : X \to Y

with rationally connected generic fibre between smooth, proper, rationally connected varieties, over

R

, and a

C^{\infty}

loop

γ : S^{1} \to X (R)

that we want to approximate, in the Euclidean topology, by a Zariski closed submanifold of

X

, assuming that we can solve the same problem on

Y

as well as on the fibres of

p

. By assumption, we can approximate

p \circ γ : S^{1} \to Y (R)

by a

C^{\infty}

map

ξ : S^{1} \to Y (R)

with Zariski closed image. The best we can hope to find, then, is a

C^{\infty}

loop

\tilde{γ} : S^{1} \to X (R)

arbitrarily close to

γ

and such that

p \circ \tilde{γ} = ξ

. We draw two conclusions:

(1) If such a

\tilde{γ}

exists, the next and final step is not finding an algebraic approximation for a

C^{\infty}

loop in a fibre of

p

, but, rather, considering the algebraic curve

B

underlying

ξ (S^{1})

, viewing

\tilde{γ}

as a

C^{\infty}

section of the projection

(X \times_{Y} B) (R) \to B (R)

and looking for an algebraic section of

X \times_{Y} B \to B

approximating

\tilde{γ}

. Thus, even when we start with just two real varieties

X

and

Y

, we need to consider one-parameter algebraic families of fibres of

p

, rather than single fibres.

(2) Consider the example where

p

is the blow-up of a surface

Y

at a real point

b

and

γ

meets

p^{- 1} (b) (R)

, transversally. Then for any

\tilde{γ}

sufficiently close to

γ

in the Euclidean topology, the loop

p \circ \tilde{γ}

has to go through

b

. Hence

ξ

has to be required to go through

b

for a loop

\tilde{γ}

as above to exist. Thus, a condition of weak approximation type must be considered in conjunction with Questions 3.1 (as was already noted by Bochnak and Kucharz [8]).

Let us now similarly contemplate a fibration argument in the context of Question 3.3 (2). We assume that

X \overset{f}{\to} B

can be factored as

X \overset{p}{\to} Y \overset{g}{\to} B

, where the variety

Y

is smooth and proper over

R

, the morphism

p

is dominant with rationally connected generic fibre, and

g

is flat. Starting from a section

s : P \to X

f

over

P

such that

{s |}_{P (R)}

can be extended to a

C^{\infty}

section

s^{'}

{f |}_{X (R)}

, a positive answer to Question 3.3 (2) for

g

produces for us a section

τ

g

that extends

p \circ s

. Let

Z = p^{- 1} (τ (B))

and let

h : Z \to B

denote the restriction of

f

. At this point, one would like to apply a positive answer to Question 3.3 (2) for

h

to obtain a section of

h

extending

s

, thus completing the argument, as

Z \subseteq X

. In order to do so, one needs to know that

{s |}_{P (R)} : P (R) \to Z (R)

can be extended to a

C^{\infty}

section of

{h |}_{Z (R)} : Z (R) \to B (R)

. However, the map

{h |}_{Z (R)}

in general even fails
to be surjective. To correct this problem, one should require, at the very least, that

τ (B (R))

approximate, in the Euclidean topology, the image of

p \circ s^{'} : B (R) \to Y (R)

. Thus, all in all, an approximation condition in the Euclidean topology has to be considered in conjunction with Question 3.3 (2).

The above discussion leads to the following definition. (This definition slightly differs from that given in [6], which considers the more general question of approximating holomorphic maps by algebraic ones, Ã la Runge, and which, as a consequence, is useful also for studying complex curves on complex varieties, without reference to the reals; however, all of the statements we make below are true with respect to either of the definitions.)

Definition 3.4. Let

B

be a smooth, proper, connected curve over R. A variety

X

over

k = R (B)

satisfies the tight approximation property if for any proper model

f : X \to B

X

over

B

with

X

smooth over

R

, any reduced 0 -dimensional subscheme

P \subset B

, any section

s^{'} : P \to X

f

over

P

and any

C^{\infty}

section

s : B (R) \to X (R)

{f |}_{X (R)}

such that

{s |}_{P (R)} = {s^{'} |}_{P (R)}

, there exists a section

σ : B \to X

f

such that

{σ |}_{P} = {s^{'} |}_{P}

and such that

{σ |}_{B (R)}

lies arbitrarily close to

s

in the compact-open topology.

Given a smooth, proper, rationally connected variety

X

over

R

, the validity of the tight approximation property for the variety obtained from

X

by extension of scalars from

R

R (t)

implies positive answers to Questions 3.1 for

X

The tight approximation property is (tautologically) a birational invariant, and it holds for

P_{k}^{n}

by a theorem of Bochnak and Kucharz [8]. (In [8], weak approximation conditions at complex points are ignored, but they create no additional difficulty.) The next two results provide more examples of varieties satisfying tight approximation.

3.3. Descent

The following theorem implements the descent method for the tight approximation property, in full generality (including non-abelian descent, as formalised by Harari and Skorobogatov). Its proof, given in [6], builds on the work of Scheiderer [94] and, in the case where

G

Theorem 3.5. Let

k

be the function field of a real curve. Let

X

be a smooth variety over

k

. Let

G

be a linear algebraic group over

k

. Let

f : Y \to X

be a left torsor under

G

. Consider twists

f^{'} : Y^{'} \to X

f

by right torsors under

G

, over

k

. If every such

Y^{'}

satisfies the tight approximation property, then so does

X

3.4. Fibration

The next theorem implements the fibration method for the tight approximation property, in full generality. Its proof, contained in [6], makes essential use of the weak toroidalisation theorem of Abramovich, Denef, and Karu [1] to establish a version of the NÃ©ron smoothening process (as in

[11, 3.1 / 3]

) for higher-dimensional bases-the point being that in the discussion at the beginning of Section 3.2, the loop

\tilde{γ}

is easily seen to exist once the morphism

p

is smooth along

γ

(see [5, LEMMA 6.11]).

Theorem 3.6. Let

k

be the function field of a real curve. Let

p : Z \to B

be a dominant morphism between smooth varieties over

k

. If

B

and the fibres of

p

above the rational points of a dense open subset of B satisfy the tight approximation property, then so does

Z

3.5. Homogeneous spaces

We are now in a position to sketch the proof of the following theorem, which in the "constant case," i.e., when the algebraic group and the homogeneous space are both defined over

R

, immediately implies Theorem B.

Theorem 3.7. Homogeneous spaces of connected linear algebraic groups over the function field of a real curve satisfy the tight approximation property.

The proof of Theorem 3.7 starts by noting that quasitrivial tori over

k

are

k

-rational, hence satisfy the tight approximation property (since so does

P_{k}^{n}

). Any torus

T

can be inserted into an exact sequence

1 \to S \to Q \to T \to 1

where

S

is a torus and

Q

is a quasitrivial torus. As any twist of

Q

as a torsor remains isomorphic to

Q

(Hilbert's Theorem 90) and hence satisfies the tight approximation property, we deduce, by the descent method (Theorem 3.5), that all tori over

k

satisfy the tight approximation property. Next, as every connected linear algebraic group over

k

is birationally equivalent to a relative torus over a

k

-rational variety (namely over the variety of maximal tori, when the algebraic group is reductive), we deduce, by the fibration method (Theorem 3.6), that connected linear algebraic groups over

k

satisfy the tight approximation property. By descent (Theorem 3.5 again), it follows that homogeneous spaces of connected linear algebraic groups over

k

satisfy the tight approximation property when they have a rational point. Finally, it is a theorem of Scheiderer that homogeneous spaces of connected linear algebraic groups over

k

satisfy the Hasse principle with respect to the real closures of

k

, so that if

X

denotes such a homogeneous space, then

X (k) \neq \emptyset

whenever a

C^{\infty}

section

s : B (R) \to X (R)

as in Definition 3.4 exists. This completes the proof of Theorem 3.7.

3.6. Further comments

Theorem 3.7 implies that homogeneous spaces of connected linear algebraic groups over the function field of a real curve satisfy weak approximation, as conjectured by ColliotThÃ©lÃ¨ne. Indeed, in the notation of Definition 3.4, if

X

is such a homogeneous space and

P

contains the locus of singular fibres of

f

, Scheiderer's work implies that

f^{- 1} (b) (R)

is nonempty and connected for all

b \in B (R) ∖ P (R)

, so that a

C^{\infty}

section

s : B (R) \to X (R)

with

{s |}_{P (R)} = {s^{'} |}_{P (R)}

always exists.

The main open problem surrounding the notion of tight approximation is the following.

Question 3.8. Let

k

be the function field of a real curve. Do all rationally connected varieties over

k

satisfy the tight approximation property?

Building on Theorems 3.5 and 3.6, the tight approximation property is shown in [6] to hold for various classes of rationally connected varieties beyond homogeneous spaces of connected linear algebraic groups. For instance, it holds for smooth cubic hypersurfaces
of dimension

\geq 2

that are defined over

R

, thus yielding, for such hypersurfaces, a positive answer to (the second part of) Question 3.1 (1).

Question 3.8 is open for cubic surfaces over

k

. Even Question 3.3 (2) is open when

X

is a cubic surface, although Question 3.3 (1) has an affirmative answer in this case, by a theorem of Tian [99].

In another direction, Question 3.8 is open for surfaces defined over

R

, and so is (the second part of) Question 3.1 (1). By inspecting the birational classification of geometrically rational surfaces and using the fibration method (Theorem 3.6), one can see that a positive answer to these questions for surfaces defined over

R

would follow from a positive answer for del Pezzo surfaces of degree 1 or 2 defined over

R

. In these cases, it would suffice, by an application of the descent method (Theorem 3.5), to know that for any real del Pezzo surface

X

of degree 1 or 2 , the universal torsors of

X

, in the sense of Colliot-ThÃ©lÃ¨ne and Sansuc [25], are R-rational whenever they have a real point. This last question, unfortunately, is very much open-even the unirationality of real del Pezzo surfaces of degree 1 is unknown. In fact, not a single example of a minimal real del Pezzo surface of degree 1 is known to be unirational. For a description of these surfaces, see [93, & 5].

Naturally, one hopes for the answer to Question 3.8 to be in the affirmative in general. This conjecture would have a host of interesting consequences, among which: a version of the Graber-Harris-Starr theorem over the reals (i.e., a positive answer to Question 3.3 (2) when

P = \emptyset

); Lang's widely open conjecture from [70] that the function field of a real curve with no real point is

C_{1}

(see [55, coRoLLARY 1.5] for the implication); and the existence of a geometrically rational curve on any smooth, proper, rationally connected variety of dimension

\geq 1

over

R

This last consequence is a conjecture of KollÃ¡r, who showed the existence of rational curves on those real rationally connected varieties of dimension

\geq 1

that have real points (see [2, Remarks 20]). For real rationally connected varieties with no real point, it is interesting to consider a weaker property: the existence of a geometrically irreducible curve of even geometric genus. The latter can be reinterpreted in terms of the real integral Hodge conjecture (see [4]). Using Hodge theory and a real adaptation of Green's infinitesimal criterion for the density of Noether-Lefschetz loci, such curves of even genus can be shown to exist on all real Fano threefolds (see [5]). However, even on smooth quartic hypersurfaces in

P_{R}^{4}

, the existence of geometrically rational curves remains a challenge, as well as the mere existence of an absolute bound, independent of the chosen quartic hypersurface, on the minimal geometric genus of a geometrically irreducible curve of even geometric genus lying on such a hypersurface.

4. FUNCTION FIELDS OF CURVES OVER $p$ -ADIC FIELDS

4.1. Some motivation: rational curves over number fields

Even though the main questions about rational points of rationally connected varieties over number fields and over function fields of real curves are still wide open, the

Brauer-Manin obstruction and the tight approximation property at least provide rather satisfactory conjectural answers. It would be highly desirable to obtain a similar conjectural picture for rational points over other fields, for significant classes of varieties-including, at a minimum, concrete criteria for the existence of rational points.

Over the field

Q (t)

, this would encompass questions about rational curves on rationally connected varieties over

Q

, about which very little is known. For example, it is unknown whether any rationally connected variety of dimension

\geq 1

over

Q

that possesses a rational point also contains a rational curve defined over

Q

. Much more ambitiously, it is unknown whether any such variety contains enough rational curves to imply the finiteness of the set of

R

-equivalence classes of rational points, a question asked in [16, QuEsTION 10.12]. (Known results on this problem are listed after Question 10.12 in [16].) As another example, the regular inverse Galois problem over

Q

, which asks for the construction of a regular Galois extension of

Q (t)

with specified Galois group, and which can be reinterpreted as a problem about the existence of appropriate rational curves on the homogeneous space

{S L}_{n} / G

over

Q

is open even for finite nilpotent groups

G

. All of these problems are currently out of reach.

As a first step towards these questions, let us replace

Q

with its completions and turn to rational points over the field

Q_{p} (t)

or over its finite extensions.

4.2. Rational curves on varieties over $p$ -adic fields

In the constant case (that is, for varieties obtained by scalar extension from varieties defined over a

p

-adic field, i.e., a finite extension of

Q_{p}

), various existence results are known:

(1) the regular inverse Galois problem over

Q_{p}

has a positive solution (first proved by Harbater [39], by "formal patching"; reproved and generalised in different directions by Pop [91] and by Colliot-ThÃ©lÃ¨ne [15]; see also [65,74,80]);

(2) for any smooth, proper, rationally connected variety

X

over a

p

-adic field

k

, KollÃ¡r

[64, 66]

has shown that the rational points of

X

fall into finitely many

R

-equivalence classes, and that there exist rational curves on

X

, defined over

k

, passing through any finite set of rational points of

X

that belong to the same

R

-equivalence class (with prescribed jets of any given order at these points).

This last statement concerns conditions of weak approximation type that can be imposed on rational curves on rationally connected varieties over

p

-adic fields. It would be interesting to formulate an analogue, in this

p

-adic context, of the surjectivity of the Borel-Haefliger cycle class map

{C H}_{1} (X) \to H_{1} (X (R), Z / 2 Z)

(i.e., of Questions 3.1 (2)).

We saw in Section 3 that in order to answer questions about homology classes of rational curves on real varieties, it can be useful to consider more generally the tight approximation property, for nonconstant varieties over the function field of a real curve. By analogy, this gives incentive to investigate the possibility of a

p

-adic analogue of the tight approximation property for nonconstant varieties over the function field of a curve over a

p

-adic field, the validity of which would have consequences for a likely easier to formulate

p

-adic integral Hodge conjecture for 1-cycles on varieties over

p

-adic fields.

4.3. Quadrics and other homogeneous spaces

In the nonconstant case, even the simplest varieties over

Q_{p} (t)

lead to difficult problems when it comes to their rational points. For instance, it is only a relatively recent theorem of Parimala and Suresh [86], for

p \neq 2

, and of Leep [72], based on work of Heath-Brown [54], for arbitrary

p

, that every projective quadric of dimension

\geq 7

over

Q_{p} (t)

possesses a rational point. (In the language of quadratic forms, "the

u

-invariant of

Q_{p} (t)

is equal to 8.") Many other articles have been devoted to local-global principles for varieties over function fields of curves over

p

-adic fields (e.g., [19-21, 23, 24,37,38,40-46,56-58,77,85,87,88,92,98]).

A patching technique was developed by Harbater, Hartmann and Krashen ("patching over fields," a successor to formal patching), and was applied to study rational points of homogeneous spaces over such fields. It was used, in [42], to give another proof of the aforementioned theorem of Parimala and Suresh, and, in [23], to establish, more generally, the local-global principle for the existence of rational points on smooth projective quadrics of dimension

\geq 1

over

Q_{p} (t)

(or over a finite extension of

Q_{p} (t)

), with respect to all discrete valuations on this field, when

p

is odd.

4.4. Reciprocity obstructions

Let

k

be a finite extension of

Q_{p} (t)

. Let

Ω

denote the set of equivalence classes of discrete valuations (of rank 1) on

k

and, for

v \in Ω

, let

k_{v}

denote the completion of

k

v

. Let

X

be an irreducible, smooth and proper variety over

k

. We embed

X (k)

diagonally into the product topological space

\prod_{v \in Ω} X (k_{v})

, which we shall also denote

X (A_{k})

(recall that

X

is proper).

We now explain how, building on the work of Bloch-Ogus and of Kato, an analogue of the Brauer-Manin obstruction can be set up in this context. These ideas, which are due to Colliot-ThÃ©lÃ¨ne, appear in print, and are put to use, in

[24, 82.3]

, in a very slightly different (equicharacteristic) situation. We refer the reader to [24, Â§ 2.3] for more details. (The "reciprocity obstructions" of

[37, § 4]

are weaker than those we discuss here.)

Our goal is thus to define, in complete generality, a closed subset

X {(A_{k})}^{r e c} \subseteq X (A_{k})

containing

X (k)

, using on the one hand a reciprocity law coming from

k

and on the other hand an analogue of the Brauer group of

X

Grothendieck's purity theorem for the Brauer group equates

Br (X)

with the unramified cohomology group

H_{n r}^{2} (X / k, Q / Z (1))

. We recall the definition of unramified cohomology: for any irreducible smooth variety

V

over a field

K

of characteristic 0 and any torsion Galois module

M

over

K

, the group

H_{n r}^{q} (V / K, M)

is the subgroup of the Galois cohomology group

H^{q} (K (V), M)

consisting of those classes whose residues along all codimension 1 points of

V

vanish. It is the unramified cohomology group

H_{n r}^{3} (X / k, Q / Z (2))

that will serve as a substitute for

Br (X)

here. (The shift in degree is explained by the fact that the field

k

has cohomological dimension 3 while number fields have virtual cohomological dimension 2.) For any field extension

K / k

, Bloch-Ogus theory provides an evaluation map

H_{n r}^{3} (X / k, Q / Z (2)) \to H^{3} (K, Q / Z (2)), α \mapsto α (x)

along any

K

-point

x

X

(see [7]).

Let

B

denote an irreducible normal proper scheme over

Z_{p}

with function field

k

. In contrast with what happens over number fields, here it is not one reciprocity law that will play a rÃ´le, but infinitely many of them: one for each closed point of

B

, for each such

B

. Namely, given any closed point

b \in B

, Kato

[61, § 1]

has constructed a complex

\begin{matrix} (4.1) & H^{3} (k, Q / Z (2)) \to ⨁_{ξ \in B_{1, b}} Br (κ (ξ)) \to Q / Z \end{matrix}

where

ξ

ranges over the set

B_{1, b}

of 1-dimensional irreducible closed subsets of

B

that contain

b

, and where

κ (ξ)

denotes the function field of

ξ

(which is either a global field of characteristic

p

or a local field of characteristic 0 ). The second arrow in (4.1) is the sum of the invariant maps from local class field theory at the finitely many places of

κ (ξ)

that lie over

b

. The first arrow of (4.1) is induced by residue maps

\partial_{v} : H^{3} (k_{v}, Q / Z (2)) \to Br (κ (ξ))

constructed by Kato in [61], where

v

denotes the discrete valuation of

k

defined by

ξ

For any

α \in H_{n r}^{3} (X / k, Q / Z (2))

, there are only finitely many 1-dimensional irreducible closed subsets

ξ

B

such that the map

X (k_{v}) \to Br (κ (ξ)), x \mapsto \partial_{v} (α (x))

does not identically vanish, if we denote by

v

the discrete valuation of

k

defined by

ξ

(see [24, PRoPoSITION 2.7 (II)] and note that for the proof given there, it is enough to assume that a dense open subset of

B

, rather than

B

itself, is a scheme over a field-an assumption satisfied here). As a consequence, it makes sense to define

X {(A_{k})}^{rec}

to be the set of

{(x_{v})}_{v \in Ω} \in X (A_{k})

such that for any irreducible normal proper scheme

B

over

Z_{p}

with function field

k

, for any closed point

b \in B

, and for any

α \in H_{n r}^{3} (X / k, Q / Z (2))

, the family

{(\partial_{v} (α (x_{v})))}_{ξ \in B_{1, b}} \in

⨁_{ξ \in B_{1, b}} Br (κ (ξ))

belongs to the kernel of the second arrow of (4.1). The fact that (4.1) is a complex immediately implies that

X (k) \subseteq X {(A_{k})}^{rec}

4.5. Sufficiency of the reciprocity obstruction

Although evidence is scarce, the answer to the following question might always be in the affirmative, as far as one knows:

Question 4.1. Let

k

be a finite extension of

Q_{p} (t)

. Let

X

be a smooth, proper, rationally connected variety over

k

. If

X {(A_{k})}^{rec} \neq \emptyset

, does it follow that

X (k) \neq \emptyset

Question 4.1 has a positive answer when

X

is a quadric and

p \neq 2

. Indeed, we recall from Section 4.3 that even

X (A_{k}) \neq \emptyset

then implies

X (k) \neq \emptyset

(see [23]). It also has a positive answer when

X

is birationally equivalent to a torsor under a torus over

k

. This follows from the work of Harari, Scheiderer, Szamuely, Tian [38, THEOREM 5.1], [97, Â§ 0.3.1] (modulo the comparison between the reciprocity obstruction defined here and the reciprocity obstruction considered in these articles; the latter is weaker, but turns out to suffice to detect rational points on torsors under tori). We note that there are examples of torsors under tori over

k

whose smooth compactifications

X

satisfy

X {(A_{k})}^{rec} = \emptyset

while

X (A_{k}) \neq \emptyset

(see [24, RemarQUE 5.10]). Positive answers to Question 4.1 are known in various other cases in which

X

is birationally equivalent to a homogeneous space of a connected linear algebraic group over

k

. For specific statements, we refer the reader to the articles quoted in Section 4.3. Question 4.1 remains open in general for smooth compactifications of torsors under connected linear alge-
braic groups over

k

, for smooth compactifications of homogeneous spaces of

{S L}_{n}

with finite stabilisers, and for conic bundle surfaces over

P_{k}^{1}

Question 4.1 focuses on the existence of rational points rather than on the density of

X (k)

X {(A_{k})}^{rec}

as the latter property is only known for projective space (see [3, Ñ‚HEoREM 1]) and hence for varieties that are rational as soon as they possess a rational point, such as quadrics. For smooth compactifications of tori, the density of

X (k)

X {(A_{k})}^{rec}

is known to hold off the set of discrete valuations of

k

whose residue field has characteristic

p

(see [37, THEOREM 5.2]; for the meaning of "off" here, see [103, DEFINITION 2.9]).

To obtain more positive answers to Question 4.1, it is natural to wish for flexible tools such as general descent theorems and fibration theorems. In the same way that introducing the tight approximation property and replacing Question 3.3 (2) with Question 3.8 was a key step to obtain a problem that behaves well with respect to fibrations into rationally connected varieties (see the discussion in Section 3.2), it is likely that in order to obtain compatibility with descent and fibrations, one will have to strengthen Question 4.1 by incorporating into it a

p

-adic analogue of the approximation condition in the Euclidean topology that appears in Definition 3.4. The main challenge, here, is to provide the correct formulation for such a

p

-adic tight approximation property.

We note that in any case, a general fibration theorem has to lie deep, as it would presumably give a direct route to the local-global principle for the existence of rational points on smooth projective quadrics over

k

(so far unknown when

p = 2

) and hence to the computation of the

u

-invariant of

k

(equal to 8 ; see Section 4.3). Indeed, in the case of conics over

k

, this local-global principle follows from Tate-Lichtenbaum duality [73]; applying a fibration theorem to a general pencil of hyperplane sections of a fixed smooth projective quadric of dimension

n \geq 2

would allow one to deduce the general case by induction on

n

4.6. Further questions

A good understanding of rational points of rationally connected varieties over function fields of curves over

p

-adic fields, be it via Question 4.1 or otherwise, should shed light on concrete test questions such as the following:

Questions 4.2. Let

p

be a prime number and

k

be a finite extension of

Q_{p} (t)

(1) Does the conjecture of MinÃ¡Ä and TÃ¢n on the vanishing of Massey products in Galois cohomology hold for

k

? (See Section 2.5 and

[78, 79]

(2) Is there an algorithm that takes as input a smooth, projective, rationally connected variety

X

over

k

and decides whether

X

has a rational point?

One might approach the first of these questions by trying to mimic [49] over

k

, which would require making progress on the arithmetic, over

k

, of homogeneous spaces of

{S L}_{n}

with finite supersolvable geometric stabilisers.

To put the second question in perspective, let us recall what is known about algorithms for deciding the existence of rational points on arbitrary varieties ("Hilbert's tenth problem") over various fields of interest. Over

Q

C (t)

, the existence of such an algorithm
is an outstanding open problem. Denef [29] showed that over

R (t)

, such an algorithm does not exist. His method was extended to prove that there is no such algorithm over

Q_{p} (t)

(Kim and Roush [62], completed by Degroote and Demeyer [27]), over any finite extension of

R (t)

that possesses a real place (Moret-Bailly [81]), or, when

p \neq 2

, over any finite extension of

Q_{p} (t)

(EisentrÃ¤ger [32], Moret-Bailly [81]). In addition, over number fields, it is known that restricting from arbitrary varieties to smooth projective varieties makes no difference (see [96, Â§ II.7], [90, THEOREM 1.1 (I)]). Restricting to smooth, projective, rationally connected varieties, however, does make a drastic difference: Question 4.2 (2) might well have an affirmative answer for all of the fields just mentioned. Over

C (t)

, this is trivially so, by the Graber-Harris-Starr theorem. Over

R (t)

, a positive answer to Question 4.2 (2) would follow from a positive answer to Question 3.8. Indeed, in the notation of Definition 3.4, if

X

satisfies the tight approximation property, then

X

has a rational point if and only if

{f |}_{X (R)}

admits a

C^{\infty}

section, a property that can be decided algorithmically. Over number fields, as was observed by Poonen [89, Remark 5.3], a positive answer to Question 4.2 (2) would follow from the conjecture that rational points are always dense in the Brauer-Manin set. It seems likely that a positive answer to Question 4.1 would similarly imply a positive answer to Question 4.2 (2). To mimic Poonen's argument, one runs into the difficulty that the elements of

H_{n r}^{3} (X / k, Q / Z (2))

are harder to describe than those of

H_{n r}^{2} (X / k, Q / Z (1)) = Br (X)

, whose interpretation in terms of Azumaya algebras is a key point in [89, REMARK 5.3]; however, this can be remedied by viewing

H_{n r}^{3} (X / k, Q / Z (2))

, using Bloch-Ogus theory, as the group of

global sections of the Zariski sheaf associated with the presheaf

U \mapsto H_{e t}^{3} (U, Q / Z (2))

, and describing

H_{e t}^{3} (U, Q / Z (2))

via ÄŒech cohomology.

4.7. Other fields

There are a number of other fields over which a better understanding of rational points of rationally connected varieties would be valuable. One of the simplest example is the fraction field

k = C ((x, y))

of the ring of formal power series

C [[x, y]]

, which can be seen as a first step before considering function fields of complex surfaces. This field presents both local and global features, and a reciprocity obstruction can again be defined (in terms of the unramified Brauer group-recall that

k

has cohomological dimension 2). This obstruction was used in [24] to produce the first example of a torsor

Y

under a torus, over

k

, such that

Y (k) = \emptyset

but

Y (k_{v}) \neq \emptyset

for every discrete valuation

v

k

. The analogues of Question 4.1 and of Questions 4.2 can be asked over this field, too. It is not known, however, whether the reciprocity obstruction explains the absence of rational points on smooth proper varieties that are birationally equivalent to torsors under tori over

k

(though see [59, CORoLLAIRE 4.4] for a closely related result involving possibly ramified Brauer classes). We refer the interested reader to

[18, 22, 59, 60]

for the state of the art.

ACKNOWLEDGMENTS

I am grateful to Jean-Louis Colliot-ThÃ©lÃ¨ne, Antoine Ducros, HÃ©lÃ¨ne Esnault, JÃ¡nos KollÃ¡r, and JÃ¡n MinÃ¡Ä for their comments on a first version of the text, and to Olivier

Benoist and Yonatan Harpaz for the pleasant collaborations that have led to the results reported on in this article.

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OLIVIER WITTENBERG

Institut GalilÃ©e, UniversitÃ© Sorbonne Paris Nord, 99 avenue Jean-Baptiste ClÃ©ment, 93430 Villetaneuse, France, wittenberg @ math.univ-paris13.fr

LIST OF CONTRIBUTORS

Aganagic, Mina 3:2108

Andreev, Nikolai 1:322

Ardila-Mantilla, Federico 6:4510

Asok, Aravind 3:2146

Bach, Francis 7:5398

Baik, Jinho 6:4190

Ball, Keith 4:3104

Bamler, Richard H. 4:2432

Bansal, Nikhil 7:5178

Bao, Gang 7:5034

Barreto, Andre 6:4800

Barrow-Green, June 7:5748

Bauerschmidt, Roland 5:3986

Bayer, Arend 3:2172

Bedrossian, Jacob 7:5618

Beliaev, Dmitry 1:V

Berger, Marsha J. 7:5056

Berman, Robert J. 4:2456

Bestvina, Mladen 2:678

Beuzart-Plessis, RaphaÃ«l 3:1712
Bhatt, Bhargav

2 : 712

Binyamini, Gal 3:1440

Blumenthal, Alex 7:5618

Bodineau, Thierry 2:750

Bonetto, Federico 5:4010

BÃ¶ttcher, Julia 6:4542

Braverman, Alexander 2:796

Braverman, Mark 1:284

Brown, Aaron 5:3388

Buckmaster, Tristan 5:3636

Burachik, Regina S. 7:5212

Burger, Martin 7:5234

Buzzard, Kevin 2:578

Calegari, Danny 4:2484

Calegari, Frank 2:610

Caprace, Pierre-Emmanuel 3:1554

Caraiani, Ana 3:1744

Cardaliaguet, Pierre 5:3660

Carlen, Eric 5:4010

Cartis, Coralia 7:5256

Chaika, Jon 5:3412

Champagnat, Nicolas 7:5656

Cieliebak, Kai 4:2504

Cohn, Henry 1:82

Colding, Tobias Holck 2:826

Collins, BenoÃ®t 4:3142

Dai, Yu-Hong 7:5290

Darmon, Henri 1:118

Dasgupta, Samit 3:1768

de la Salle, Mikael 4:3166

De Lellis, Camillo 2:872

Delarue, FranÃ§ois 5:3660

Delecroix, Vincent 3:2196

Demers, Mark F. 5:3432

Ding, Jian 6:4212

Dobrinen, Natasha 3:1462

Dong, Bin 7:5420

Drivas, Theodore D. 5:3636

Du, Xiumin 4:3190

Dujardin, Romain 5:3460

Duminil-Copin, Hugo 1:164

Dwork, Cynthia 6:4740

Dyatlov, Semyon 5:3704

E, Weinan 2:914

Efimov, Alexander I. 3:2212

Eldan, Ronen 6:4246

Etheridge, Alison 6:4272

Fasel, Jean 3:2146

Feigin, Evgeny 4:2930

Ferreira, Rita 5:3724

Fisher, David 5:3484

Fonseca, Irene 5:3724
Fournais, SÃ¸ren 5:4026

Frank, Rupert L. 1:142, 5:3756

Friedgut, Ehud 6:4568

Funaki, Tadahisa 6:4302

Gallagher, Isabelle 2:750

Gamburd, Alexander 3:1800

Gentry, Craig 2:956

Georgieva, Penka 4:2530

Giuliani, Alessandro 5:4040

GonÃ§alves, PatrÃcia 6:4326

Gotlib, Roy 6:4842

Goujard, Ã‰lise 3:2196

Gould, Nicholas I. M. 7:5256

Grima, Clara I. 7:5702

Guionnet, Alice 2:1008

Gupta, Neena 3:1578

Guth, Larry 2:1054

Gwynne, Ewain 6:4212

Habegger, Philipp 3:1838

Hairer, Martin 1:26

Hastings, Matthew B. 5:4074

Hausel, TamÃ¡s 3:2228

Helmuth, Tyler 5:3986

Hesthaven, Jan S. 7:5072

Higham, Nicholas J. 7:5098

Hintz, Peter 5:3924

Holden, Helge 1:11

Holzegel, Gustav 5:3924

Hom, Jennifer 4:2740

Houdayer, Cyril 4:3202

Huh, June 1:212

Ichino, Atsushi 3:1870

Imhausen, Annette 7:5772

Ionescu, Alexandru D. 5:3776

Iritani, Hiroshi 4:2552

Isaksen, Daniel C. 4:2768

Jackson, Allyn 1:548, 1:554

$1 : 560, 1 : 566$

Jain, Aayush 6:4762

Jegelka, Stefanie 7:5450

Jia, Hao 5:3776

Jitomirskaya, Svetlana 2:1090

Kakde, Mahesh 3:1768

Kalai, Gil 1:50

Kaletha, Tasho 4:2948

Kamnitzer, Joel 4:2976

Kang, Hyeonbae 7:5680

Kato, Syu 3:1600

Kaufman, Tali 6:4842

Kazhdan, David 2:796

Kenig, Carlos 1:5, 1:9

Kleiner, Bruce 4:2376

Klingler, Bruno 3:2250

Knutson, Allen 6:4582

Koukoulopoulos, Dimitris 3:1894

Kozlowski, Karol Kajetan 5:4096

Krichever, Igor 2:1122

Kutyniok, Gitta 7:5118

Kuznetsov, Alexander 2:1154

Lacoin, Hubert 6:4350

Larsen, Michael J. 3:1624

LemaÅ„czyk, Mariusz 5:3508

Lepski, Oleg V. 7:5478

LeVeque, Randall J. 7:5056

Levine, Marc 3:2048

Lewin, Mathieu 5:3800

Li, Chi 3:2286
Lin, Huijia 6:4762

Liu, Gang 4:2576

Liu, Yi 4:2792

Loeffler, David 3:1918

Loss, Michael 5:4010

LÃ¼, Qi 7:5314

Lugosi, GÃ¡bor 7:5500

Luk, Jonathan 5:4120

MacrÃ¬, Emanuele 3:2172

Mann, Kathryn 4:2594

Marks, Andrew S. 3:1488

Maynard, James 1:240

McLean, Mark 4:2616

Mikhailov, Roman 4:2806

Mohammadi, Amir 5:3530

Mossel, Elchanan 6:4170

Nakanishi, Kenji 5:3822

Nazarov, Alexander I. 5:3842

Neeman, Amnon 3:1636

Nelson, Jelani 6:4872

Nickl, Richard 7:5516

Nikolaus, Thomas 4:2826

Norin, Sergey 6:4606

Novik, Isabella 6:4622

Novikov, Dmitry 3:1440

Ogata, Yoshiko 5:4142

Okounkov, Andrei 1:376, 1:414

1 : 460, 1 : 492

Ozdaglar, Asuman 7:5340

Pagliantini, Cecilia 7:5072

Panchenko, Dmitry 6:4376

Paternain, Gabriel P. 7:5516

Peeva, Irena 3:1660

Perelman, Galina 5:3854

Pierce, Lillian B. 3:1940

Pixton, Aaron 3:2312

Pramanik, Malabika 4:3224

Pretorius, Frans 2:652

Procesi, Michela 5:3552

Prokhorov, Yuri 3:2324

Punshon-Smith, Sam 7:5618

Ramanan, Kavita 6:4394

Ramasubramanian, Krishnamurthi 7:5784

Randal-Williams, Oscar 4:2856

Rasmussen, Jacob 4:2880

Raz, Ran 1:106

Regev, Oded 6:4898

Remenik, Daniel 6:4426

Ripamonti, NicolÃ² 7:5072

Safra, Muli (Shmuel) 6:4914

Sahai, Amit 6:4762

Saint-Raymond, Laure 2:750

Sakellaridis, Yiannis 4:2998

Saloff-Coste, Laurent 6:4452

Sayin, Muhammed O. 7:5340

Schacht, Mathias 6:4646

Schechtman, Gideon 4:3250

SchÃ¶lkopf, Bernhard 7:5540

Schwartz, Richard Evan 4:2392

Scott, Alex 6:4660

Sfard, Anna 7:5716

Shan, Peng 4:3038

Shapira, Asaf 6:4682

Sheffield, Scott 2:1202

Shin, Sug Woo 3:1966

Shkoller, Steve 5:3636
Shmerkin, Pablo 4:3266

Silver, David 6:4800

Silverman, Joseph H. 3:1682

Simonella, Sergio 2:750

Smirnov, Stanislav 1:V

Solovej, Jan Philip 5:4026

Soundararajan, Kannan 1:66, 2:1260

Stroppel, Catharina 2:1312

Sturmfels, Bernd 6:4820

Sun, Binyong 4:3062

Svensson, Ola 6:4970

Taimanov, Iskander A. 4:2638

Tarantello, Gabriella 5:3880

Tian, Ye 3:1990

Tikhomirov, Konstantin 4:3292

Toint, Philippe L. 7:5296

Tokieda, Tadashi 1:160

Tran, Viet Chi 7:5656

Tucsnak, Marius 7:5374

Ulcigrai, Corinna 5:3576

Van den Bergh, Michel 2:1354

Venkatraman, Raghavendra 5:3724

Viazovska, Maryna 1:270

Vicol, Vlad 5:3636

Vidick, Thomas 6:4996

von KÃ¼gelgen, Julius 7:5540

Wahl, Nathalie 4:2904

Wang, Guozhen 4:2768

Wang, Lu 4:2656

Wang, Weiqiang 4:3080

Ward, Rachel 7:5140

Wei, Dongyi 5:3902

Weiss, Barak 5:3412

White, Stuart 4:3314

Wigderson, Avi 2:1392

Williams, Lauren K. 6:4710

Willis, George A. 3:1554

Wittenberg, Olivier 3:2346

Wood, Melanie Matchett 6:4476

Xu, Zhouli 4:2768

Ying, Lexing 7:5154

Yokoyama, Keita 3:1504
Young, Robert J. 4:2678

Zerbes, Sarah Livia 3:1918

Zhang, Cun-Hui 7:5594

Zhang, Kaiqing 7:5340

Zhang, Zhifei 5:3902

Zheng, Tianyi 4:3340

Zhou, Xin 4:2696

Zhu, Chen-Bo 4:3062

Zhu, Xiaohua 4:2718

Zhu, Xinwen 3:2012

Zhuk, Dmitriy 3:1530

Zograf, Peter 3:2196

Zorich, Anton 3:2196

P RES S

2 We encode $T$ , e.g., by its recursive index.

3 In [32], Paris showed that ${I t P H}_{2}^{3}$ is independent of $P A$ , while his argument implies the equivalence of statements 2 and 3. See Section 5.2.
5 See Section 5 for the construction of the moduli space $M_{d}^{N}$ .
6 Although in fairness it should be noted that [91] suggests the opposite conclusion, stating: "Are there any rational periodic orbits of a quadratic $x^{2} + c$ of period greater than 3? The results for periods 1,2 , and 3 would lead one to suspect that there must be."
7 We remark that it is easy to prove uniform boundedness for $x^{d} + c$ over $Q$ when $d$ is odd, and more generally over any field $K / Q$ with a real embedding. Indeed, it is an elementary fact that if $f : R \to R$ is any nondecreasing function, then $f$ has no nonfixed periodic points; cf. [64].

8 The gonality of an algebraic curve $X$ , or its function field, is the minimal degree of a nonconstant map $X \to P^{1}$ .
1 Of course, the following discussion also applies to $U (W)$ .

2 A Whittaker datum of $U (V^{'})$ is a pair $(N, θ)$ consisting of a maximal unipotent subgroup

$N \subset U (V^{'})$ and a generic character $θ : N \to C^{\times}$ . This datum only matters up to conjugacy.
1 We make a small abuse of notation by using $X$ to denote both the conjugacy class from the introduction, which is used in the definition of a Shimura datum, and the quotient $G (R) / K_{\infty}^{\circ} A_{\infty}^{\circ}$ considered in this section. See $[24 $ § 2.4]$ for an extended discussion of the various quotients.
3 As a consequence of the comparison with moduli spaces of local shtukas in [56], one obtains a group-theoretic characterization of Rapoport-Zink spaces as local Shimura varieties determined by the tuple $(G, b, μ)$ . We suppress $(G, μ)$ from the notation for simplicity.
4 The result precedes the notion of diamonds and, in order to ensure that $S_{K^{p}}^{\circ b}$ is a diamond, one needs to take care in defining it. At hyperspecial level, one should consider the adic generic fiber of the formal completion of the integral model of the Shimura variety along the Newton stratum indexed by $b$ in its special fiber.
2 See [125] and [81] for definition and properties of expanders.

3 Let $p$ be a large prime and denote by $X^{*} (p) = X (p) ∖ (0, 0, 0)$ the solutions of (1.1) modulo $p$ with the removal of $(0, 0, 0)$ . The Markoff graphs are obtained by joining each $x$ in $X^{*} (p)$ to $R_{j} (x), j = 1, 2, 3$ . They were considered first by Arthur Baragar in his thesis [3].
6 The exponent $\frac{1}{3}$ in (1.5) has been improved to $\frac{7}{9}$ in [87].

7 We remark that in [52] Corvaja and Zannier showed that the greatest prime factor of $x y$ for a Markoff triple $(x, y, z)$ tends to infinity.
8 "Trotz der auÃŸerordentlich merkwÃ¼rdigen und wichtigen Resultate scheinen diese schwiergen Untersuchungen wenig bekannt zu sein" [In spite of the extraordinarily noteworthy and important results these difficult investigations seem to be little known]
2 More precisely, we obtain a relation to the first derivative $L^{'} (V_{π}^{*}, 1)$ , with $V_{π}$ being 1critical. Unfortunately, this computation does not give us any information about the Ã©tale class in $H^{1} (E, V_{π})$ , since the motivic class might be in the kernel of the Ã©tale realisation map. (This is the fundamental obstruction to proving the Bloch-Kato conjecture for 1critical Galois representations.)
3 This argument can be used to construct an Eichler-Shimura isomorphism in families for ${G S p}_{4}$ , which interpolates the classical $H^{1}$ comparison isomorphism at almost all classical points - see [51].
3 Sometimes these methods extend beyond the stated hypotheses. For example, [18] also works for Kisin-Pappas models (Section 6), and [26] proves a uniformization result also in the nonbasic case. However, we will not try to present the methods in their maximally general settings.
2 If char $F =$ char $k$ (the equal characteristic case), this assumption on $k$ is not necessary. We impose it here to have a uniform treatment of both equal and mixed characteristic (i.e., char $F \neq char k)$ cases. For the same reason, we work with perfect algebraic geometry below even in equal characteristic.
5 Such restriction functor defines the so-called fusion product, a key concept in the geometric Satake equivalence. The terminology "fusion" originally comes from conformal field theory.
1 Voevodsky showed this is not possible integrally, so the best one can hope for is a $t$ structure with $Q$ -coefficients.
2 In fact, at the beginning of $§ 3$ of [28], Bloch and Kato write, "The cohomological symbol defined by Tate [114] gives a map . . . which one conjectures to be an isomorphism quite generally."
1 For example, the square-tiled surface in Figure 1 is made up of 54 squares, has 3 conical points of angle $3 π$ (corresponding to simple zeros of $q$ ), and 7 conical points of angle $π$ (corresponding to simple poles of $q$ ). Therefore, it has genus 0 and belongs to the principal stratum $Q (1^{3}, - 1^{7})$ .
1 In the classical formulation, Tian's CM weight or, equivalently, the Donaldson-Futaki invariant is used to define the K-stability. However, to fit our discussion in the nonArchimedean framework, we use the equivalent formulation via the $M^{N A}$ functional.

2.2. Non-Archimedean pluripotential theory

2.3. Stability of Fano varieties

2.4. Normalized volume and local stability theory of klt singularities

3. ARCHIMEDEAN (COMPLEX ANALYTIC) THEORY VS. NON-ARCHIMEDEAN THEORY

3.1. Correspondence between Archimedean and non-Archimedean objects

3.2. Yau-Tian-Donaldson conjecture for general polarized manifolds

3.3. YTD conjecture for Fano varieties

3.3.1. Non-Archimedean approach

3.3.2. Other approaches

ACKNOWLEDGMENTS

FUNDING

REFERENCES

CHI LI

THE DOUBLE RAMIFICATION CYCLE FORMULA

AARON PIXTON

ABSTRACT

MATHEMATICS SUBJECT CLASSIFICATION 2020

KEYWORDS

1. INTRODUCTION

2. TAUTOLOGICAL CLASSES

2.1. Preliminaries

2.2. Stable graphs

2.3. Compact type

3. PREVIOUS FORMULAS AND RESULTS

4. MOTIVATION FOR THE FORMULA

4.1. Expanding Hain's formula

4.2. Cohomological field theory axioms

4.3. Givental's R-matrix action

4.4. Divergent averages

4.5. Interpolating finite rank CohFTs

4.6. Geometric interpretation from ( k / r ) ( k / r ) (k//r)(k / r)(k/r)-spin structures

5. THE DOUBLE RAMIFICATION CYCLE FORMULA

ACKNOWLEDGMENTS

FUNDING

REFERENCES

AARON PIXTON

EFFECTIVE RESULTS IN THE THREE-DIMENSIONAL MINIMAL MODEL PROGRAM YURI PROKHOROV

ABSTRACT

MATHEMATICS SUBJECT CLASSIFICATION 2020

KEYWORDS

Describe all the intermediate steps and the outcome of the MMP.

1. SINGULARITIES

2. MINIMAL MODEL PROGRAM

3. GENERAL ELEPHANT

4. DIVISORIAL CONTRACTIONS TO A POINT

5. DEL PEZZO FIBRATIONS

6. EXTREMAL CURVE GERMS

6.1. Construction of germs by deformations

7. EXTREMAL CURVE GERMS: GENERAL ELEPHANT

7.1. Local method

7.2. Extension from S ∈ | − 2 K X | S âˆˆ âˆ’ 2 K X S in|-2K_(X)|S \in\left|-2 K_{X}\right|Sâˆˆ|âˆ’2KX|

7.3. Global method

8. SEMISTABLE GERMS

9. EXCEPTIONAL CURVE GERMS

10. Q Q Q\mathbb{Q}Q-FANO THREEFOLDS

ACKNOWLEDGMENTS

FUNDING

REFERENCES

YURI PROKHOROV

SOME ASPECTS OF RATIONAL POINTS AND RATIONAL CURVES

OLIVIER WITTENBERG

ABSTRACT

MATHEMATICS SUBJECT CLASSIFICATION 2020

KEYWORDS

1. INTRODUCTION

2. SOLVABLE GROUPS AND THE GRUNWALD PROBLEM IN INVERSE GALOIS THEORY

2.1. Homogeneous spaces

2.2. Geometry

2.3. Descent

2.4. Fibration

2.5. An application to Massey products

3. RATIONAL CURVES ON REAL ALGEBRAIC VARIETIES

3.1. A few questions

3.2. Tight approximation

3.3. Descent

3.4. Fibration

3.5. Homogeneous spaces

3.6. Further comments

4. FUNCTION FIELDS OF CURVES OVER p p ppp-ADIC FIELDS

4.1. Some motivation: rational curves over number fields

4.6. Geometric interpretation from $(k / r)$ -spin structures

EFFECTIVE RESULTS IN THE THREE-DIMENSIONAL MINIMAL MODEL PROGRAM
YURI PROKHOROV

7.2. Extension from $S \in | - 2 K_{X} |$

10. $Q$ -FANO THREEFOLDS

4. FUNCTION FIELDS OF CURVES OVER $p$ -ADIC FIELDS

4.2. Rational curves on varieties over $p$ -adic fields

$1 : 560, 1 : 566$